Hindawi Publishing CorporationJournal of OptimizationVolume 2013 Article ID 297015 12 pageshttpdxdoiorg1011552013297015

Research ArticleNondifferentiable Minimax Programming Problems in ComplexSpaces Involving Generalized Convex Functions

Anurag Jayswal Ashish Kumar Prasad and Krishna Kummari

Department of Applied Mathematics Indian School of Mines Dhanbad Jharkhand 826 004 India

Correspondence should be addressed to Anurag Jayswal anurag jais123yahoocom

Received 17 June 2013 Accepted 7 November 2013

Academic Editor Sheng-Jie Li

Copyright copy 2013 Anurag Jayswal et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We start our discussion with a class of nondifferentiable minimax programming problems in complex space and establish sufficientoptimality conditions under generalized convexity assumptions Furthermore we derive weak strong and strict converse dualitytheorems for the two types of dual models in order to prove that the primal and dual problems will have no duality gap under theframework of generalized convexity for complex functions

1 Introduction

The literature of the mathematical programming is crowdedwith necessary and sufficient conditions for a point to bean optimal solution to the optimization problem Levinson[1] was the first to study mathematical programming incomplex space who extended the basic theorems of linearprogramming over complex space In particular using avariant of the Farkas lemma from real space to complexspace he generalized duality theorems from real linearprogramming Since then linear fractional nonlinear andnonlinear fractional complex programming problems werestudied by many researchers (see [2ndash5])

Minimax problems are encountered in several importantcontexts One of the major context is zero sum games wherethe objective of the first player is to minimize the amountgiven to the other player and the objective of the secondplayer is to maximize this amount Ahmad and Husain [6]established sufficient optimality conditions for a class of non-differentiable minimax fractional programming problemsinvolving (119865 120572 120588 119889)-convexity Later on Jayswal et al [7]extended the work of Ahmad and Husain [6] to establishsufficient optimality conditions and duality theorems forthe nondifferentiable minimax fractional problem under theassumptions of generalized (119865 120572 120588 119889)-convexity RecentlyJayswal and Kumar [8] established sufficient optimalityconditions and duality theorems for a class of nondiffer-entiable minimax fractional programming problems under

the assumptions of (119862 120572 120588 119889)-convexity Lai et al [9] estab-lished several sufficient optimality conditions for minimaxprogramming in complex spaces under the assumptions ofgeneralized convexity of complex functions Subsequentlythey applied the optimality conditions to formulate paramet-ric dual and derived weak strong and strict converse dualitytheorems

The first work on fractional programming in complexspace appeared in 1970 when Swarup and Sharma [10]generalized the results of Charnes and Cooper [11] to thecomplex space Lai and Huang [12] showed that a minimaxfractional programming problem is equivalent to a minimaxnonfractional parametric problem for a given parameter incomplex space and established the necessary and sufficientoptimality conditions for nondifferentiable minimax frac-tional programming problem with complex variables undergeneralized convexity assumptions

Recently Lai and Liu [13] considered a nondifferentiableminimax programming problem in complex space and estab-lished the appropriate duality theorems for parametric dualand parameter free dual models They showed that there isno duality gap between the two dual problems with respectto the primal problem under some generalized convexities ofcomplex functions in the complex programming problem

In this paper we focus our study on nondifferentiableminimax programming over complex spaces The paper isorganized as follows In Section 2 we recall some notations

2 Journal of Optimization

and definitions in complex spaces In Section 3 we establishsufficient optimality conditions under generalized convexityassumptions Weak strong and strict converse duality the-orems related to nondifferentiable minimax programmingproblems in complex spaces for two types of dual models areestablished in Sections 4 and 5 followed by the conclusion inSection 6

2 Notations and Preliminaries

We use the following notations that appear in most works onmathematical programming in complex space

119862119899(119877119899) = 119899-dimensional vector space of complex

(real) numbers119862119898times119899(119877119898times119899)= the set of119898times119899 complex (real)matrices

119877119899+

= 119909 isin 119877119899 119909119895

ge 0 119895 = 1 2 119899 = thenonnegative orthant of 119877119899

119860119867 = 119860119879 = the conjugate transpose of 119860 = [119886

119894119895]

⟨119911 119906⟩ = 119906119867119911 = the inner product of 119906 119911 in 119862119899

Now we recall some definitions related to mathematicalprogramming in complex space that are used in the sequel ofthe paper

Definition 1 (see [5]) A subset 119878 sube 119862119899 is polyhedral cone if

there is 119896 isin 119873 and 119860 isin 119862119899times119896 such that 119878 = 119860119877119896+

= 119860119909 | 119909 isin

119877119896+ that is 119878 is generated by a finite number of vectors (the

columns of 119860)Equivalently 119878 sube 119862

119899 is said to be a polyhedral cone ifit is the intersection of a finite number of closed half-spaceshaving the origin on the boundary that is there is a naturalnumber 119901 and 119901-points 119906

1 1199062 119906

119901such that

119878 =

119901

⋂119896=1

119863(119906119896) = 119911 isin 119862

119899

| Re ⟨119911 119906119896⟩ ge 0 119896 = 1 2 119901

(1)

where 119863(119906119896) 119896 = 1 2 119901 are closed half-spaces involving

the point 119906119896

Definition 2 (see [5]) If 0 = 119878 sub 119862119899 then 119878lowast = 119910 isin 119862119899 |

for all 119911 isin 119878 rArr Re(119910119867119911) ge 0 constitute the dual (polar) of119878

IfΘ 119862119899 rarr 119862 is analytic in a neighbourhood of 119911

0isin 119862119899

thennabla119911Θ(1199110) = [120597Θ(119911

0)120597119911119894] 119894 = 1 2 119899 is the gradient of

functionΘ at 1199110 Similarly if the complex functionΘ(1199081 1199082)

is analytic in 2119899 variables (1199081 1199082) and (1199110 1199110) isin 1198622119899 we

define the gradients by

nabla119911Θ(1199110 1199110) = [

120597Θ (1199110 1199110)

1205971199081119895

] 119895 = 1 2 119899

nabla119911Θ(1199110 1199110) = [

120597Θ (1199110 1199110)

1205971199082119895

] 119895 = 1 2 119899

(2)

In this paper we consider the following complex program-ming problem

min120577isin119883

sup120578isin119884

Re [119891 (120577 120578) + (119911119867119860119911)12

]

subject to 120577 isin 119883 = 120577 = (119911 119911) isin 1198622119899 | minusℎ (120577) isin 119878

(P)

where 119884 = 120578 = (119908119908) | 119908 isin 119862119898 is a compact subset in1198622119898 119860 isin 119862119899times119899 is a positive semidefinite Hermitian matrix 119878is a polyhedral cone in 119862119901 119891(sdot sdot) is continuous and for each120578 isin 119884 119891(sdot 120578) 1198622119899 rarr 119862 and ℎ(sdot) 1198622119899 rarr 119862119901 are analytic in119876 = (119911 119911) | 119911 isin 119862119899 sub 1198622119899 where119876 is a linear manifold overa real field In order to have a convex real part for a nonlinearanalytic function the complex functions need to be definedon the linear manifold over 119877 that is 119876 = 120577 = (119911 119911) isin 1198622119899 |

119911 isin 119862119899

Special Cases (i) If problem (P) is a real programming prob-lem with two variables nondifferentiable minimax problemit may be expressed as

min sup119910isin119884

119891 (119909 119910) + (119909119879119861119909)12

st 119892 (119909) le 0 119909 isin 119877119899

(3)

where 119884 is compact subset of 119877119897 119891(sdot sdot) 119877119899 times 119877119897 rarr 119877 and119892(sdot) 119877119899 rarr 119877119898 are continuously differentiable functions at119909 isin 119877119899 and 119861 is a positive semidefinite symmetric matrixThis problem was studied by Ahmad et al [14 15]

(ii) If 119884 vanishes in (P) then problem (P) reduces to theproblem considered by Mond and Craven [16] that is

min Re [119891 (120577) + (119911119867119860119911)

12

]

st 120577 isin 119883 = 120577 isin 1198622119899 | minusℎ (120577) isin 119878 120577 = (119911 119911) 119911 isin 119862119899

(P1)

(iii) If 119860 = 0 then (P) becomes a differentiable complexminimax programming problem studied by Datta and Bhatia[3] that is

min120577isin119883

sup120578isin119884

Re119891 (120577 120578)

st 120577 isin 119883 = 120577 isin 1198622119899 | minusℎ (120577) isin 119878 (P0)

Definition 3 A functional 119865 119862119899 times 119862119899 times 119862119899 rarr 119877 is said tobe sublinear in its third variable if for any 119911

1 1199112

isin 119862119899 thefollowing conditions are satisfied

(i) 119865(1199111 1199112 1199061+ 1199062) le 119865(119911

1 1199112 1199061) + 119865(119911

1 1199112 1199062)

(ii) 119865(1199111 1199112 120572119906) = 120572119865(119911

1 1199112 119906)

for any 120572 ge 0 in 119877+and 119906

1 1199062 119906 isin 119862119899 From (ii) it is clear

that 119865(1199111 1199112 0) = 0

Let 119865 119862119899 times 119862119899 times 119862119899 rarr 119877 be sublinear on the thirdvariable 120579 119862119899 times 119862119899 rarr 119877

+with 120579(119911

1 1199112) = 0 if 119911

1= 1199112and

120572 119862119899 times 119862119899times rarr 119877+ 0 Let 119891 and ℎ be analytic functions

and 120588 let be a real number Now we introduce the followingdefinitions which are extensions of the definitions given byLai et al [9] and Mishra and Rueda [17]

Journal of Optimization 3

Definition 4 The real part Re[119891] of analytic function 119891 119876 sub

1198622119899 rarr 119862 is said to be (119865 120572 120588 120579)-convex (strict (119865 120572 120588 120579)-convex) with respect to 119877

+on the manifold 119876 = 120577 = (119911 119911) |

119911 isin 119862119899 sub 1198622119899 if for any 120577 = (119911 119911) 1205770= (1199110 1199110) isin 119876 one has

Re [119891 (119911 119911) minus 119891 (1199110 1199110)]

ge (gt) 119865 (119911 1199110 120572 (119911 119911

0)

times (nabla119911119891 (1199110 1199110) + nabla119911119891 (1199110 1199110))) + 120588120579

2

(119911 1199110)

(4)

Definition 5 The real part Re[119891] of analytic function 119891

119876 sub 1198622119899 rarr 119862 is said to be (119865 120572 120588 120579)-quasiconvex (strict(119865 120572 120588 120579)-quasiconvex) with respect to 119877

+on the manifold

119876 = 120577 = (119911 119911) | 119911 isin 119862119899 sub 1198622119899 if for any 120577 = (119911 119911)1205770= (1199110 1199110) isin 119876 one has

Re [119891 (119911 119911) minus 119891 (1199110 1199110)]

le (lt) 0 997904rArr 119865(119911 1199110 120572 (119911 119911

0)

times (nabla119911119891 (1199110 1199110) + nabla119911119891 (1199110 1199110)))

le minus1205881205792

(119911 1199110)

(5)

Definition 6 The real part Re[119891] of analytic function 119891

119876 sub 1198622119899 rarr 119862 is said to be (119865 120572 120588 120579)-pseudoconvex (strict(119865 120572 120588 120579)-pseudoconvex) with respect to119877

+on themanifold

119876 = 120577 = (119911 119911) | 119911 isin 119862119899 sub 1198622119899 if for any 120577 = (119911 119911)1205770= (1199110 1199110) isin 119876 one has

119865 (119911 1199110 120572 (119911 119911

0) (nabla119911119891 (1199110 1199110) + nabla119911119891 (1199110 1199110)))

ge minus1205881205792

(119911 1199110) 997904rArr Re [119891 (119911 119911) minus 119891 (119911

0 1199110)]

ge (gt) 0

(6)

Definition 7 The mapping ℎ 1198622119899 rarr 119862119901 is said to be(119865 120572 120588 120579)-convex (strict (119865 120572 120588 120579)-convex) with respect tothe polyhedral cone 119878 sub 119862

119901 on the manifold 119876 if for any120583 isin 119878 and 120577 = (119911 119911) 120577

0= (1199110 1199110) isin 119876 one has

Re ⟨120583 ℎ (120577) minus ℎ (1205770)⟩

ge (gt) 119865 (119911 1199110 120572 (119911 119911

0)

times (120583119879

nabla119911ℎ (1205770) + 120583119867

nabla119911ℎ (1205770))) + 120588120579

2

(119911 1199110)

(7)

Definition 8 The mapping ℎ 1198622119899 rarr 119862119901 is said to be(119865 120572 120588 120579)-quasiconvex (strict (119865 120572 120588 120579)-quasiconvex) with

respect to the polyhedral cone 119878 sub 119862119901 on the manifold 119876 if

for any 120583 isin 119878 and 120577 = (119911 119911) 1205770= (1199110 1199110) isin 119876 one has

Re ⟨120583 ℎ (120577) minus ℎ (1205770)⟩

le (lt) 0 997904rArr 119865(119911 1199110 120572 (119911 119911

0)

times (120583119879

nabla119911ℎ (1205770) + 120583119867

nabla119911ℎ (1205770)))

le minus1205881205792

(119911 1199110)

(8)

Definition 9 The mapping ℎ 1198622119899 rarr 119862119901 is said to be(119865 120572 120588 120579)-pseudoconvex (strict (119865 120572 120588 120579)-pseudoconvex)with respect to the polyhedral cone 119878 sub 119862

119901 on the manifold119876 if for any 120583 isin 119878 and 120577 = (119911 119911) 120577

0= (1199110 1199110) isin 119876 one has

119865(119911 1199110 120572 (119911 119911

0) (120583119879

nabla119911ℎ (1205770) + 120583119867

nabla119911ℎ (1205770)))

ge 1205881205792

(119911 1199110) 997904rArr Re ⟨120583 ℎ (120577) minus ℎ (120577

0)⟩

ge (gt) 0

(9)

Remark 10 In the proofs of theorems sometimes it may bemore convenient to use certain alternative but equivalentforms of the above definitions Consider the following exam-ple

The real part Re[119891] of analytic function119891 119876 sub 1198622119899 rarr 119862

is said to be (119865 120572 120588 120579)-pseudoconvex with respect to 119877+on

the manifold 119876 = 120577 = (119911 119911) | 119911 isin 119862119899 sub 1198622119899 if for any1205770= (1199110 1199110) isin 119876 one has

Re [119891 (119911 119911) minus 119891 (1199110 1199110)]

lt 0 997904rArr 119865(119911 1199110 120572 (119911 119911

0)

times (nabla119911119891 (1199110 1199110) + nabla119911119891 (1199110 1199110)))

lt minus1205881205792

(119911 1199110)

(10)

Remark 11 If we take 120572(119911 1199110) = 1 then the above definitions

reduce to that given by Lai et al [9] In addition if we take120588 = 0 then we obtain the definitions given by Mishra andRueda [17]

Let 119860 isin 119862119899times119899 and 119911 119906 isin 119862119899 then Schwarz inequality can

be written as

Re (119911119867119860119906) le (119911119867

119860119911)12

(119906119867

119860119906)12

(11)

The equality holds if 119860119911 = 120582119860119906 or 119911 = 120582119906 for 120582 ge 0

Definition 12 (see [12]) The problem (P) is said to satisfy theconstraint qualification at a point 120577

0= (1199110 1199110) if for any

nonzero 120583 isin 119878lowast sub 119862119901

Re ⟨ℎ1015840120577(1205770) (120577 minus 120577

0) 120583⟩ = 0 for 120577 = 120577

0 (12)

In the next section we recall some notations and discussnecessary and sufficient optimality conditions for problem(P) on the basis of Lai and Liu [18] and Lai and Huang [12]

4 Journal of Optimization

3 Necessary and Sufficient Conditions

Let 119891(120577 sdot) 120577 = (119911 119911) isin 1198622119899 be a continuous function definedon119884 where119884 sub 1198622119898 is a specified compact subset in problem(P) Then the supremum sup]isin119884 Re119891(120577 ]) will be attained toits maximum in 119884 and the set

119884 (120577) = 120578 isin 119884 | Re119891 (120577 120578) = sup]isin119884

Re119891 (120577 ]) (13)

is then also a compact set in 1198622119898 In particular if 120577 = 1205770

=

(1199110 1199110) is an optimal solution of problem (P) there exist a

positive integer 119896 and finite points 120578119894isin 119884(120577

0) 120582119894gt 0 119894 =

1 2 119896 withsum119896

119894=1120582119894= 1 such that the Lagrangian function

120601 (120577) =

119896

sum119894=1

120582119894119891 (120577 120578

119894) + ⟨ℎ (120577) 120583⟩ (120583 = 0 in 119878

lowast

) (14)

satisfies the Kuhn-Tucker type condition at 1205770 That is

(

119896

sum119894=1

1205821198941198911015840

120577(1205770 120578119894) + ⟨ℎ

1015840

120577(1205770) 120583⟩) (120577 minus 120577

0) = 0 (15)

Re ⟨ℎ (1205770) 120583⟩ = 0 (16)

Equivalent form of expression (15) at 120577 = 1205770isin 119876 is

119896

sum119894=1

120582119894[nabla119911119891 (1205770 120578119894) + nabla119911119891 (1205770 120578119894)]

+ (120583119879

nabla119911ℎ (1205770) + 120583119867

nabla119911ℎ (1205770)) = 0

(17)

For the integer 119896 corresponding a vector 120578 equiv (1205781 1205782 120578

119896) isin

119884(1205770)119896 and 120582

119894gt 0 119894 = 1 2 119896 withsum

119896

119894=1120582119894= 1 we define a

set as follows

119885120578(1205770) =

120577 isin 1198622119899

| minusℎ1015840

120577(1205770) 120577 isin 119878 (minusℎ (120577

0))

120577 = (119911 119911) isin 119876

Re[119896

sum119894=1

1205821198941198911015840

120577(1205770 120578119894) 120577 + ⟨119860119911 119911⟩

12

] lt 0

(18)

where the set 119878(1199040) is the intersection of closed half-spaces

having the point 1199040isin 119878 on their boundaries

Theorem 13 (necessary optimality conditions) Let 1205770

=

(1199110 1199110) isin 119876 be an optimal solution to (P) Suppose that the

constraint qualification is satisfied for (P) at 1205770and 119911119867

01198601199110

=

⟨1198601199110 1199110⟩ gt 0 Then there exist 0 = 120583 isin 119878lowast sub 119862119901 119906 isin 119862119899 and a

positive integer 119896 with the following properties

(i) 120578119894isin 119884(1205770) 119894 = 1 2 119896

(ii) 120582119894gt 0 119894 = 1 2 119896 sum119896

119894=1120582119894= 1

such that sum119896

119894=1120582119894119891(120577 120578

119894) + ⟨ℎ(120577) 120583⟩ + ⟨119860119911 119911⟩

12 satisfies thefollowing conditions

119896

sum119894=1

120582119894nabla119911119891 (1205770 120578119894) + nabla119911119891 (1205770 120578119894) + 119860119906

+ (120583119879

nabla119911ℎ (1205770) + 120583119867

nabla119911ℎ (1205770)) = 0

(19)

Re ⟨ℎ (1205770) 120583⟩ = 0 (20)

119906119867

119860119906 le 1 (21)

(119911119867

01198601199110)12

= Re (1199111198670119860119906) (22)

Theorem 14 (sufficient optimality conditions) Let 1205770

=

(1199110 1199110) isin 119876 be a feasible solution to (P) Suppose that there

exists a positive integer 119896 120582119894gt 0 120578119894isin 119884(1205770) 119894 = 1 2 119896with

sum119896

119894=1120582119894= 1 and 0 = 120583 isin 119878lowast sub 119862119901 satisfying conditions (19)ndash

(22) Further if Re[sum119896119894=1

120582119894[119891(120577 120578

119894) + 119911119867119860119906]] is (119865 120572

1 1205881 120579)-

convex with respect to 119877+on 119876 ℎ(120577) is (119865 120572

2 1205882 120579)-convex on

119876with respect to the polyhedral cone 119878 sub 119862119901 and12058811205721(119911 1199110)+

12058821205722(119911 1199110) ge 0 then 120577

0= (1199110 1199110) is an optimal solution to

(P)

Proof We prove this theorem by contradiction Suppose thatthere is a feasible solution 120577 isin 119876 such that

sup120578isin119884

Re [119891 (120577 120578) + (119911119867

119860119911)12

]

lt sup120578isin119884

Re [119891 (1205770 120578) + (119911

119867

01198601199110)12

]

(23)

Since 120578119894isin 119884(1205770) 119894 = 1 2 119896 we have

sup120578isin119884

Re [119891 (1205770 120578) + (119911

119867

01198601199110)12

]

= Re [119891 (1205770 120578119894) + (119911

119867

01198601199110)12

]

for 119894 = 1 2 119896

Re [119891 (120577 120578119894) + (119911

119867

119860119911)12

]

lt sup120578isin119884

Re [119891 (120577 120578) + (119911119867

119860119911)12

]

for 119894 = 1 2 119896

(24)

Thus from the above three inequalities we obtain

Re [119891 (120577 120578119894) + (119911

119867

119860119911)12

]

lt Re [119891 (1205770 120578119894) + (119911

119867

01198601199110)12

]

for 119894 = 1 2 119896

(25)

Journal of Optimization 5

Using (21) and generalized Schwarz inequality we get

Re (119911119867119860119906) le (119911119867

119860119911)12

(119906119867

119860119906)12

le (119911119867

119860119911)12

= Re [(119911119867119860119911)12

]

(26)

and inequality (22) yields

Re (1199111198670119860119906) = Re [(119911119867

01198601199110)12

] (27)

Using (26) and (27) in (25) we have

Re [119891 (120577 120578119894) + 119911119867

119860119906] lt Re [119891 (1205770 120578119894) + 119911119867

0119860119906]

for 119894 = 1 2 119896

(28)

Since 120582119894gt 0 and sum

119896

119894=1120582119894= 1 we have

119903119897Re[119896

sum119894=1

120582119894[119891 (120577 120578

119894) + 119911119867

119860119906] minus

119896

sum119894=1

120582119894[119891 (1205770 120578119894) + 119911119867

0119860119906]]

lt 0

(29)

Since Re[sum119896119894=1

120582119894[119891(120577 120578

119894) + 119911119867119860119906]] is (119865 120572

1 1205881 120579)-convex

with respect to 119877+on 119876 we have

Re[119896

sum119894=1

120582119894[119891 (120577 120578

119894) + 119911119867

119860119906]

minus

119896

sum119894=1

120582119894[119891 (1205770 120578119894) + 119911119867

0119860119906]]

ge 119865[119911 1199110 1205721(119911 1199110)

times(

119896

sum119894=1

120582119894nabla119911119891 (1205770 120578119894) + nabla119911119891 (1205770 120578119894) + 119860119906)]

+ 12058811205792

(119911 1199110)

(30)

From (29) and (30) we conclude that

119865[119911 1199110 1205721(119911 1199110)

times(

119896

sum119894=1

120582119894nabla119911119891(1205770 120578119894) + nabla119911119891 (1205770 120578119894) + 119860119906)]

lt minus12058811205792

(119911 1199110)

(31)

which due to sublinearity of 119865 can be written as

119865[119911 1199110

119896

sum119894=1

120582119894nabla119911119891 (1205770 120578119894) + nabla119911119891 (1205770 120578119894) + 119860119906]

lt minus1205881

1205721(119911 1199110)1205792

(119911 1199110)

(32)

On the other hand from the feasibility of 120577 to (P) we haveminusℎ(120577) isin 119878 or Re⟨ℎ(120577) 120583⟩ le 0 for 120583 isin 119878lowast which along with(20) yields

Re ⟨ℎ (120577) 120583⟩ le 0 = Re ⟨ℎ (1205770) 120583⟩ (33)

Since ℎ(120577) is (119865 1205722 1205882 120579)-convex on 119876 with respect to the

polyhedral cone 119878 sub 119862119901 we have

Re ⟨ℎ (120577) 120583⟩ minus Re ⟨ℎ (1205770) 120583⟩

ge 119865 [119911 1199110 1205722(119911 1199110) (120583119879

nabla119911ℎ (1205770) + 120583119867

nabla119911ℎ (1205770))]

+ 12058821205792

(119911 1199110)

(34)

From (33) and (34) it follows that

119865 [119911 1199110 1205722(119911 1199110) (120583119879

nabla119911ℎ (1205770) + 120583119867

nabla119911ℎ (1205770))]

le minus12058821205792

(119911 1199110)

(35)

which due to sublinearity of 119865 can be written as

119865 [119911 1199110 120583119879

nabla119911ℎ (1205770) + 120583119867

nabla119911ℎ (1205770)] le minus

1205882

1205722(119911 1199110)1205792

(119911 1199110)

(36)

On adding (32) and (36) and using sublinearity of 119865 we get

119865[119911 1199110

119896

sum119894=1

120582119894nabla119911119891 (1205770 120578119894) + nabla119911119891 (1205770 120578119894) + 119860119906

+120583119879

nabla119911ℎ (1205770) + 120583119867

nabla119911ℎ (1205770) ]

lt minus(1205881

1205721(119911 1199110)+

1205882

1205722(119911 1199110)) 1205792

(119911 1199110)

(37)

The above inequality together with the assumption12058811205721(119911 1199110) + 12058821205722(119911 1199110) ge 0 gives

119865[119911 1199110

119896

sum119894=1

120582119894nabla119911119891 (1205770 120578119894) + nabla119911119891 (1205770 120578119894) + 119860119906

+ 120583119879

nabla119911ℎ (1205770) + 120583119867

nabla119911ℎ (1205770) ] lt 0

(38)

which contradicts (19) hence the theorem

6 Journal of Optimization

Theorem 15 (sufficient optimality conditions) Let 1205770

=

(1199110 1199110) isin 119876 be a feasible solution to (P) Suppose that there

exists a positive integer 119896 120582119894gt 0 120578119894isin 119884(1205770) 119894 = 1 2 119896with

sum119896

119894=1120582119894= 1 and 0 = 120583 isin 119878lowast sub 119862119901 satisfying conditions (19)ndash

(22) Further if Re[sum119896119894=1

120582119894[119891(120577 120578

119894) + 119911119867119860119906]] is (119865 120572

1 1205881 120579)-

pseudoconvex with respect to 119877+on 119876 ℎ(120577) is (119865 120572

2 1205882 120579)-

quasiconvex on 119876 with respect to the polyhedral cone 119878 sub 119862119901and 120588

11205721(119911 1199110) + 12058821205722(119911 1199110) ge 0 then 120577

0= (1199110 1199110) is an

optimal solution to (P)

Proof Proceeding as in Theorem 14 we have

Re[119896

sum119894=1

120582119894[119891 (120577 120578

119894) + 119911119867

119860119906] minus

119896

sum119894=1

120582119894[119891 (1205770 120578119894) + 119911119867

0119860119906]]

lt 0

(39)

which by (119865 1205721 1205881 120579)-pseudoconvexity of

Re[sum119896119894=1

120582119894[119891(120577 120578

119894) + 119911119867119860119906]] with respect to 119877

+on 119876

yields

119865[119911 1199110 1205721(119911 1199110)

119896

sum119894=1

120582119894nabla119911119891 (1205770 120578119894) + nabla119911119891 (1205770 120578119894) + 119860119906]

lt minus12058811205792 (119911 119911

0)

(40)

Using the sublinearity of 119865 the above inequality can bewritten as

119865[119911 1199110

119896

sum119894=1

120582119894nabla119911119891 (1205770 120578119894) + nabla119911119891 (1205770 120578119894) + 119860119906]

lt minus1205881

1205721(119911 1199110)1205792 (119911 119911

0)

(41)

On the other hand from the feasibility of 120577 to (P) we haveminusℎ(120577) isin 119878 or Re⟨ℎ(120577) 120583⟩ le 0 for 120583 isin 119878lowast which along with(20) yields

Re ⟨ℎ (120577) 120583⟩ le 0 = Re ⟨ℎ (1205770) 120583⟩ (42)

Since ℎ(120577) is (119865 1205722 1205882 120579)-quasiconvex on 119876 with respect to

the polyhedral cone 119878 sub 119862119901 the above inequality yields

119865 [119911 1199110 1205722(119911 1199110) (120583119879

nabla119911ℎ (1205770) + 120583119867

nabla119911ℎ (1205770))]

le minus12058821205792 (119911 119911

0)

(43)

which due to sublinearity of 119865 can be written as

119865 [119911 1199110 120583119879

nabla119911ℎ (1205770) + 120583

119867

nabla119911ℎ (1205770)] le minus

1205882

1205722(119911 1199110)1205792

(119911 1199110)

(44)

On adding (41) and (44) and using sublinearity of 119865 we get

119865[119911 1199110

119896

sum119894=1

120582119894nabla119911119891 (1205770 120578119894) + nabla119911119891 (1205770 120578119894) + 119860119906

+ 120583119879

nabla119911ℎ (1205770) + 120583119867

nabla119911ℎ (1205770) ]

lt minus(1205881

1205721(119911 1199110)+

1205882

1205722(119911 1199110)) 1205792

(119911 1199110)

(45)

The above inequality together with the assumption12058811205721(119911 1199110) + 12058821205722(119911 1199110) ge 0 gives

119865[119911 1199110

119896

sum119894=1

120582119894nabla119911119891 (1205770 120578119894) + nabla119911119891 (1205770 120578119894) + 119860119906

+120583119879

nabla119911ℎ (1205770) + 120583119867

nabla119911ℎ (1205770) ] lt 0

(46)

which contradicts (19) hence the theorem

Theorem 16 (sufficient optimality conditions) Let 1205770

=

(1199110 1199110) isin 119876 be a feasible solution to (P) Suppose that there

exists a positive integer 119896 120582119894gt 0 120578119894isin 119884(1205770) 119894 = 1 2 119896with

sum119896

119894=1120582119894= 1 and 0 = 120583 isin 119878lowast sub 119862119901 satisfying conditions (19)ndash

(22) Further if Re[sum119896119894=1

120582119894[119891(120577 120578

119894) + 119911119867119860119906]] is (119865 120572

1 1205881 120579)-

quasiconvex with respect to 119877+on119876 ℎ(120577) is strict (119865 120572

2 1205882 120579)-

pseudoconvex on119876 with respect to the polyhedral cone 119878 sub 119862119901

and 12058811205721(119911 1199110) + 12058821205722(119911 1199110) ge 0 then 120577

0= (1199110 1199110) is an

optimal solution to (P)

Proof The proof follows on the similar lines of Theorem 15

4 Parametric Duality

We adopt the following notations in order to simplify theformulation of dual

119870 (120585) = (119896 120578) isin 119873 times 119877119896

+times 1198622119898119896

|

= (1205821 1205822 120582

119896) with

119896

sum119894=1

120582119894= 1

120578 = (1205781 1205782 120578

119896)

with 120578119894isin 119884 (120585) 119894 = 1 2 119896

(47)

for 120585 = (119906 119906) isin 119876 sub 1198622119899Now we formulate a parametric dual problem (D1) with

respect to the complex minimax programming problem (P)as follows

max(119896120582120578)isin119870(120585)

sup(120585120583119908119905)isin119883(119896120582120578)

119905(D1)

Journal of Optimization 7

where119883(119896 120578) denotes the set of all (120585 120583 119908 119905) isin 1198622119899 times119862119901 times

119862119899 times 119877 to satisfy the following conditions119896

sum119894=1

120582119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]

+ 120583119879 nabla119911ℎ (120585) + 120583119867nabla

119911ℎ (120585) = 0

(48)

119896

sum119894=1

120582119894Re119891 (120585 120578

119894) + (119906

119867

119860119906)12

minus 119905 ge 0 (49)

Re ⟨ℎ (120585) 120583⟩ ge 0 (50)

119908119867

119860119908 le 1 (51)

(119906119867

119860119906)12

= Re (119906119867119860119908) (52)

0 = 120583 isin 119878lowast

(53)

If for a triplet (119896 120578) isin 119870(120585) the set 119883(119896 120578) = 0then we define the supremum over 119883(119896 120578) to be minusinfin fornonexecption in the formulation of (D1)

Theorem 17 (weak duality) Let 120577 = (119911 119911) and(119896 120578 120585 120583 119908 119905) be feasible solutions to (P) and (D1)respectively Further if Re[sum119896

119894=1120582119894[119891(120577 120578

119894) + 119911119867119860119908]] is

(119865 1205721 1205881 120579)-pseudoconvex with respect to 119877

+on 119876 ℎ(120577) is

(119865 1205722 1205882 120579)-quasiconvex on 119876 with respect to the polyhedral

cone 119878 sub 119862119901 and 12058811205721(119911 119906) + 120588

21205722(119911 119906) ge 0 Then

sup120578isin119884

Re [119891 (120577 120578) + (119911119867

119860119911)12

] ge 119905 (54)

Proof Suppose on the contrary that

sup120578isin119884

Re [119891 (120577 120578) + (119911119867

119860119911)12

] lt 119905 (55)

By compactness of 119884(120585) sub 119884 in 119862119901 120585 isin (119906 119906) isin 119876 thereexist an integer 119896 gt 0 and finite points 120578

119894isin 119884(120585) 120582

119894gt 0

119894 = 1 2 119896 with sum119896

119894=1120582119894= 1 such that (49) holds From

(49) and (55) we have

Re[119896

sum119894=1

120582119894[119891 (120577 120578

119894) + (119911

119867

119860119911)12

]] lt

119896

sum119894=1

120582119894119905

le

119896

sum119894=1

120582119894Re [119891 (120585 120578

119894) + (119906

119867

119860119906)12

]

(56)

From (51) and the generalized Schwarz inequality we have

Re (119911119867119860119908) le (119911119867

119860119911)12

(119908119867

119860119908)12

le (119911119867

119860119911)12

(57)

Using (52) and (57) in (56) we get

Re[119896

sum119894=1

120582119894[119891 (120577 120578

119894) + 119911119867

119860119908]]

lt Re[119896

sum119894=1

120582119894[119891 (120585 120578

119894) + 119906119867

119860119908]]

(58)

Since Re[sum119896119894=1

120582119894[119891(120577 120578

119894) + 119911119867119860119908]] is (119865 120572

1 1205881 120579)-

pseudoconvex with respect to 119877+on 119876 the above inequality

implies that

119865[119911 119906 1205721(119911 119906)

119896

sum119894=1

120582119894nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]

lt minus12058811205792

(119911 119906)

(59)

which due to sublinearity of 119865 can be written as

119865[119911 119906

119896

sum119894=1

120582119894nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]

lt minus1205881

1205721(119911 119906)

1205792

(119911 119906)

(60)

By the feasibility of 120577 = (119911 119911) to (P) we have minusℎ(120577) isin 119878 orRe⟨ℎ(120577) 120583⟩ le 0 for 120583 isin 119878lowast which along with (50) yields

Re ⟨ℎ (120577) 120583⟩ le 0 le Re ⟨ℎ (120585) 120583⟩ (61)

The above inequality together with the (119865 1205722 1205882 120579)-

quasiconvexity of ℎ(120577) on 119876 with respect to the polyhedralcone 119878 sub 119862

119901 implies

119865 [119911 119906 1205722(119911 119906) (120583

119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

le minus12058821205792

(119911 119906)

(62)

which due to sublinearity of 119865 can be written as

119865 [119911 119906 120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585)]

le minus1205882

1205722(119911 119906)

1205792

(119911 119906) (63)

On adding (60) and (63) and using sublinearity of 119865 we get

119865[119911 119906

119896

sum119894=1

120582119894nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908

+ 120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585) ]

lt minus(1205881

1205721(119911 119906)

+1205882

1205722(119911 119906)

) 1205792

(119911 119906)

(64)

From the assumption 12058811205721(119911 119906) +120588

21205722(119911 119906) ge 0 the above

inequality yields

119865[119911 119906

119896

sum119894=1

120582119894nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908

+ 120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585) ] lt 0

(65)

which contradicts (48) hence the theorem

8 Journal of Optimization

Theorem 18 (weak duality) Let 120577 = (119911 119911) and(119896 120578 120585 120583 119908 119905) be feasible solutions to (P) and (D1)respectively Further if Re[sum119896

119894=1120582119894[119891(120577 120578

119894) + 119911

119867

119860119908]] is(119865 1205721 1205881 120579)-quasiconvex with respect to 119877

+on 119876 ℎ(120577) is

(119865 1205722 1205882 120579)-pseudoconvex on 119876 with respect to the polyhedral

cone 119878 sub 119862119901 and 12058811205721(119911 119906) + 120588

21205722(119911 119906) ge 0 then

sup120578isin119884

Re [119891 (120577 120578) + (119911119867

119860119911)12

] ge 119905 (66)

Proof Theproof follows the same lines as inTheorem 17

Theorem 19 (strong duality) Let 1205770= (1199110 1199110) be an optimal

solution to the problem (P) at which a constraint qualificationis satisfied Then there exist (119896 120578) isin 119870(120577

0) and (120577

0 120583 119908 119905) isin

119883(119896 120578) such that (119896 120578 1205770 120583 119908 119905) is a feasible solution to

the dual problem (D1) If the hypotheses of Theorem 17 or 18are satisfied then (119896 120578 120577

0 120583 119908 119905) is optimal to (D1) and

the two problems (P) and (D1) have the same optimal values

Proof Theproof follows along the lines ofTheorem6 (Lai andLiu [13])

Theorem 20 (strict converse duality) Let 120577 and(

120582 120578 120585 120583 119908 ) be optimal solutions to (P) and (D1)

respectively and assume that the assumptions of Theorem 19are satisfied Further assume that the following conditions aresatisfied

(i) Resum119896

119894=1119894[119891(120577 120578

119894) + 119867119860119908] is strict (119865 120572

1 1205881 120579)-

pseudoconvex with respect to 119877+on 119876 and ℎ(120577) is

(119865 1205722 1205882 120579)-quasiconvex on 119876 with respect to the

polyhedral cone 119878 sub 119862119901(ii) 12058811205721( ) + 120588

21205722( ) ge 0

Then 120577 = 120585 that is 120577 is optimal solution to (D1)

Proof On the contrary suppose that ( ) = 120577 = 120585 = ( )On applyingTheorem 19 we know that

= supisin119884

Re [119891 (120577 120578) + (119867

119860)12

] (67)

From the feasibility of 120577 isin 119876 to (P) 120583 isin 119878lowast and (50) we have

Re ⟨ℎ (120577) 120583⟩ le 0 le Re ⟨ℎ (120585) 120583⟩ (68)

Since ℎ(120577) is (119865 1205722 1205882 120579)-quasiconvex on 119876 with respect to

the polyhedral cone Sin 119862119901 the above inequality yields

119865 [ 1205722( ) (120583

119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

le minus12058821205792

( )

(69)

which by sublinearity of 119865 implies

119865 [ (120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

le minus1205882

1205722( )

1205792

( )

(70)

By (48) and the sublinearity of 119865 we have

119865[

[

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]]

]

+ 119865 [ (120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

ge 119865[

[

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]

+ 120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585)]

]

= 0

(71)

The above inequality together with (70) and 12058811205721( ) +

12058821205722( ) ge 0 gives

119865[

[

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]]

]

ge minus119865 [ (120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

ge1205882

1205722( )

1205792

( )

ge minus1205881

1205721( )

1205792

( )

(72)

That is

119865[

[

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]]

]

ge minus1205881

1205721( )

1205792

( )

(73)

which by sublinearity of 119865 implies

119865[

[

1205721( )

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]]

]

ge minus12058811205792

( )

(74)

Since Resum119896

119894=1119894[119891(120577 120578

119894) + 119867119860119908] is strict (119865 120572

1 1205881 120579)-

pseudoconvex with respect to 119877+on 119876 the above inequality

implies that

Re[

[

119896

sum119894=1

119894[119891 (120577 120578

119894) + 119867

119860119908]]

]

gt Re[

[

119896

sum119894=1

119894[119891 (120585 120578

119894) + 119867

119860119908]]

]

(75)

Journal of Optimization 9

From (51) (52) and the generalized Schwarz inequality wehave

Re (119867119860119908) le (119867

119860)12

Re (119867119860119908) = (119867

119860)12

(76)

which on substituting in (75) and by using (49) we obtain

Re[

[

119896

sum119894=1

119894[119891 (120577 120578

119894) + (

119867

119860)12

]]

]

gt Re[

[

119896

sum119894=1

119894[119891 (120585 120578

119894) + (

119867

119860)12

]]

]

ge

119896

sum119894=1

119894

(77)

Consequently there exist certain 1198940which satisfy

Re [119891 (120577 1205781198940) + (

119867

119860)12

] gt (78)

Hence

supisin119884

Re [119891 (120577 120578) + (119867

119860)12

]

ge Re [119891 (120577 1205781198940) + (

119867

119860)12

] gt

(79)

which contradicts (67) hence the theorem

5 Parameter Free Duality

Making use of the optimality conditions we show thatthe following formation is a dual (D2) to the complexprogramming problem (P)

max(119896120582120578)isin119870(120585)

sup(120585120583119908)isin119883(119896120582120578)

Re [119891 (120585 120578) + (119906119867

119860119906)12

] (D2)

where119883(119896 120578) denotes the set of all (120585 120583 119908) isin 1198622119899times119862119901times119862119899

to satisfy the following conditions

119896

sum119894=1

120582119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]

+ 120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585) = 0

(80)

Re ⟨ℎ (120585) 120583⟩ ge 0 (81)

119908119867

119860119908 le 1 (82)

(119906119867

119860119906)12

= Re (119906119867119860119908) (83)

If for a triplet (119896 120578) isin 119870(120585) the set 119883(119896 120578) = 0then we define the supremum over 119883(119896 120578) to be minusinfin fornonexception in the formulation of (D2)

Nowwe establish appropriate duality theorems and provethat optimal values of (P) and (D2) are equal under theassumption of generalized convexity in order to show that theproblems (P) and (D2) have no duality gap

Theorem21 (weak duality) Let 120577 = (119911 119911) and (119896 120578 120585 120583 119908)

be feasible solutions to (P) and (D2) respectively Furtherif Re[sum119896

119894=1120582119894[119891(120577 120578

119894) + 119911119867119860119908]] is (119865 120572

1 1205881 120579)-pseudoconvex

with respect to 119877+on 119876 ℎ(120577) is (119865 120572

2 1205882 120579)-quasiconvex on 119876

with respect to the polyhedral cone 119878 sub 119862119901 and 12058811205721(z 119906) +

12058821205722(119911 119906) ge 0 then

sup120578isin119884

Re [119891 (120577 120578) + (119911119867

119860119911)12

]

ge sup120578isin119884

Re [119891 (120585 120578) + (119906119867

119860119906)12

]

(84)

Proof On the contrary we suppose that

sup120578isin119884

Re [119891 (120577 120578) + (119911119867

119860119911)12

]

lt sup120578isin119884

Re [119891 (120585 120578) + (119906119867

119860119906)12

]

(85)

Since 120578119894isin 119884(120585) sub 119884 119894 = 1 2 119896 we have

sup120578isin119884

Re [119891 (120585 120578) + (119906119867

119860119906)12

]

= Re [119891 (120585 120578119894) + (119906

119867

119860119906)12

] 119894 = 1 2 119896

(86)

Re [119891 (120577 120578119894) + (119911

119867

119860119911)12

]

le sup120578isin119884

Re [119891 (120577 120578) + (119911119867

119860119911)12

] 119894 = 1 2 119896

(87)

Then the above three inequalities give

Re [119891 (120577 120578119894) + (119911

119867

119860119911)12

]

lt Re [119891 (120585 120578119894) + (119906

119867

119860119906)12

] 119894 = 1 2 119896

(88)

From (82) (83) (88) and the generalized Schwarz inequalitywe have

Re [119891 (120577 120578119894) + (119911

119867

119860119908)]

lt Re [119891 (120585 120578119894) + (119906

119867

119860119908)] 119894 = 1 2 119896

(89)

As 120582119894gt 0 119894 = 1 2 119896 and sum

119896

119894=1120582119894= 1 we have

Re[119896

sum119894=1

120582119894[119891 (120577 120578

119894) + (119911

119867

119860119908)]]

minus Re[119896

sum119894=1

120582119894[119891 (120585 120578

119894) + (119906

119867

119860119908)]] lt 0

(90)

10 Journal of Optimization

Since Re[sum119896119894=1

120582119894[119891(120577 120578

119894) + 119911119867119860119908]] is (119865 120572

1 1205881 120579)-

pseudoconvex with respect to 119877+on 119876 the above inequality

implies that

119865[119911 119906 1205721(119911 119906)

times

119896

sum119894=1

120582119894nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]

lt minus12058811205792

(119911 119906)

(91)

which by sublinearity of 119865 becomes

119865[119911 119906

119896

sum119894=1

120582119894nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]

lt minus1205881

1205721(119911 119906)

1205792

(119911 119906)

(92)

By the feasibility of 120577 = (119911 119911) to (P) 0 = 120583 isin 119878lowast and theinequality (81) we obtain

Re ⟨ℎ (120577) 120583⟩ le 0 le Re ⟨ℎ (120585) 120583⟩ (93)

The above inequality together with the (119865 1205722 1205882 120579)-

quasiconvexity of ℎ(120577) on 119876 with respect to the polyhedralcone 119878 sub 119862

119901 implies

119865 [119911 119906 1205722(119911 119906) (120583

119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

le minus12058821205792

(119911 119906)

(94)

which by sublinearity of 119865 becomes

119865 [119911 119906 120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585)]

le minus1205882

1205722(119911 119906)

1205792

(119911 119906) (95)

On adding (92) and (95) and using the sublinearity of 119865 weget

119865[119911 119906

119896

sum119894=1

120582119894nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908

+120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585) ]

lt minus(1205881

1205721(119911 119906)

+1205882

1205722(119911 119906)

) 1205792

(119911 119906)

(96)

From the assumption 12058811205721(119911 119906) +120588

21205722(119911 119906) ge 0 the above

inequality yields

119865[119911 119906

119896

sum119894=1

120582119894nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908

+ 120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585) ] lt 0

(97)

which contradicts (80) hence the theorem

Theorem 22 (weak duality) Let 120577 = (119911 119911) and(119896 120578 120585 120583 119908) be feasible solutions to (P) and (D2)respectively Further if Re[sum119896

119894=1120582119894[119891(120577 120578

119894) + 119911

119867

119860119908]] is(119865 1205721 1205881 120579)-quasiconvex with respect to 119877

+on 119876 ℎ(120577) is

(119865 1205722 1205882 120579)-pseudoconvex on 119876 with respect to the polyhedral

cone 119878 sub 119862119901 and 12058811205721(119911 119906) + 120588

21205722(119911 119906) ge 0 then

sup120578isin119884

Re [119891 (120577 120578) + (119911119867

119860119911)12

]

ge sup120578isin119884

Re [119891 (120585 120578) + (119906119867

119860119906)12

]

(98)

Proof Theproof follows the same lines as inTheorem 21

Theorem 23 (strong duality) Let 1205770= (1199110 1199110) be an optimal

solution to the problem (P) at which a constraint qualificationis a satisfied Then there exist (119896 120578) isin 119870(120577

0) and (120577

0 120583 119908) isin

119883(119896 120578) such that (119896 120578 1205770 120583 119908) is a feasible solution to the

dual problem (D2) Further if the hypotheses of Theorem 21or Theorem 22 are satisfied then (119896 120578 120577

0 120583 119908) is optimal

to (D2) and the two problems (P) and (D2) have the sameoptimal values

Proof Theproof follows along the lines ofTheorem8 (Lai andLiu [13])

Theorem 24 (strict converse duality) Let 120577 and(

120582 120578 120585 120583 119908) be optimal solutions to (P) and (D2)

respectively and the conditions of Theorem 23 are satisfiedFurther assume that the following conditions are satisfied

(i) Resum119896

119894=1119894[119891(120577 120578

119894) + 119911119867119860119908] is strict (119865 120572

1 1205881 120579)-

pseudoconvex with respect to 119877+on 119876 and ℎ(120577) is

(119865 1205722 1205882 120579)-quasiconvex on 119876 with respect to the

polyhedral cone 119878 sub 119862119901(ii) 12058811205721( ) + 120588

21205722( ) ge 0

Then 120577 = 120585 that is 120585 is an optimal solution to (D2)

Proof On the contrary we assume that ( ) = 120577 = 120585 = ( )On applyingTheorem 23 we know that

supisin119884

Re [119891 (120577 120578) + (119867

119860)12

]

= supisin119884

Re [119891 (120585 120578) + (119867

119860)12

]

(99)

From the feasibility of 120577 isin 119876 to (P) 120583 isin 119878lowast inequality (81)yields

Re ⟨ℎ (120577) 120583⟩ le 0 le Re ⟨ℎ (120585) 120583⟩ (100)

Since ℎ(120577) is (119865 1205722 1205882 120579)-quasiconvex on 119876 with respect to

the polyhedral cone 119878 in 119862119901 the above inequality yields

119865 [ 1205722( ) (120583

119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

le minus12058821205792

( )

(101)

Journal of Optimization 11

which by sublinearity of 119865 implies

119865 [ (120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

le minus1205882

1205722( )

1205792

( )

(102)

By (80) and the sublinearity of 119865 we have

119865[

[

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]]

]

+ 119865 [ (120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

ge 119865[

[

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]

+ 120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585)]

]

= 0

(103)

The above inequality together with (102) and 12058811205721( ) +

12058821205722( ) ge 0 gives

119865[

[

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]]

]

ge minus119865 [ (120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

ge1205882

1205722( )

1205792

( )

ge minus1205881

1205721( )

1205792

( )

(104)

That is

119865[

[

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]]

]

ge minus1205881

1205721( )

1205792 ( )

(105)

which by sublinearity of 119865 implies

119865[

[

1205721( )

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]]

]

ge minus12058811205792 ( )

(106)

Since Resum119896

119894=1119894[119891(120577 120578

119894) + 119911119867119860119908] is strict (119865 120572

1 1205881 120579)-

pseudoconvex with respect to 119877+on 119876 the above inequality

implies that

Re[

[

119896

sum119894=1

119894[119891 (120577 120578

119894) + 119867

119860119908]]

]

gt Re[

[

119896

sum119894=1

119894[119891 (120585 120578

119894) + 119867

119860119908]]

]

(107)

From (82) (83) and the generalized Schwarz inequality weget

Re (119867119860119908) le (119867

119860)12

Re (119867119860119908)12

= (119867

119860)12

(108)

which on substituting in (107) we obtain

Re[

[

119896

sum119894=1

119894[119891 (120577 120578

119894) + (

119867

119860)12

] ]

]

gt Re[

[

119896

sum119894=1

119894[119891 (120585 120578

119894) + (

119867

119860)12

]]

]

(109)

Consequently there exist certain 1198940which satisfy

Re [119891 (120577 1205781198940) + (119867119860)

12

]

gt Re [119891 (120585 1205781198940) + (119867119860)

12

]

(110)

Hence

supisin119884

Re [119891 (120577 120578) + (119867

119860)12

]

ge Re [119891 (120577 1205781198940) + (

119867

119860)12

]

gt Re [119891 (120585 1205781198940) + (

119867

119860)12

]

= supisin119884

Re [119891 (120585 120578) + (119867

119860)12

]

(111)

which contradicts (99) hence the theorem

6 Conclusion

In this paper we introduced generalized (119865 120572 120588 120579)-convexfunctions and established sufficient optimality conditions fora class of nondifferentiable minimax programming problemsin complex space These optimality conditions are thenused to construct two types of dual model and finally wederived weak strong and strict converse duality theorems

12 Journal of Optimization

to show that there is no duality gap between the two dualproblems with respect to the primal problem under somegeneralized convexities of complex functions in the complexprogramming problem As a future task the authors wouldlike to extend these results to second and higher order casesand establish the relations between primal and its second andhigher order dual problems

Acknowledgments

The authors are thankful to the anonymous referees for theirvaluable comments which have improved the presentationof the paper The research of the first author is financiallysupported by the University Grant Commission New DelhiIndia through Grant No F no 41-8012012(SR)

References

[1] N Levinson ldquoLinear programming in complex spacerdquo Journalof Mathematical Analysis and Applications vol 14 no 1 pp 44ndash62 1966

[2] J C Chen and H C Lai ldquoOptimality conditions for minimaxprogramming of analytic functionsrdquoTaiwanese Journal ofMath-ematics vol 8 no 4 pp 673ndash686 2004

[3] N Datta and D Bhatia ldquoDuality for a class of nondifferentiablemathematical programming problems in complex spacerdquo Jour-nal of Mathematical Analysis and Applications vol 101 no 1 pp1ndash11 1984

[4] O P Jain and P C Saxena ldquoA duality theorem for a specialclass of programming problems in complex spacerdquo Journal ofOptimization Theory and Applications vol 16 no 3-4 pp 207ndash220 1975

[5] I M Stancu-Minasian Fractional Programming Theory Meth-ods and Applications vol 409 Kluwer Academic DordrechtGermany 1997

[6] I Ahmad and Z Husain ldquoOptimality conditions and dualityin nondifferentiable minimax fractional programming withgeneralized convexityrdquo Journal of Optimization Theory andApplications vol 129 no 2 pp 255ndash275 2006

[7] A Jayswal I Ahmad and D Kumar ldquoNondifferentiable mini-max fractional programming problem with nonsmooth gener-alized convex functionsrdquoCommunications onAppliedNonlinearAnalysis vol 18 no 4 pp 57ndash75 2011

[8] A Jayswal and D Kumar ldquoOn nondifferentiable minimaxfractional programming involving generalized (C 120572 120588 d)-convexityrdquo Communications on Applied Nonlinear Analysis vol18 no 1 pp 61ndash77 2011

[9] H-C Lai J-C Lee and S-C Ho ldquoParametric duality on mini-max programming involving generalized convexity in complexspacerdquo Journal of Mathematical Analysis and Applications vol323 no 2 pp 1104ndash1115 2006

[10] K Swarup and I C Sharma ldquoProgramming with linear frac-tional functionals in complex spacerdquo Cahiers Center drsquoEtudesRecherche Operationelle vol 12 pp 103ndash109 1970

[11] A Charnes and W W Cooper ldquoProgramming with linearfractional functionalsrdquo Naval Research Logistics Quarterly vol9 no 3-4 pp 181ndash186 1962

[12] H-C Lai and T-Y Huang ldquoOptimality conditions for non-differentiable minimax fractional programming with complexvariablesrdquo Journal of Mathematical Analysis and Applicationsvol 359 no 1 pp 229ndash239 2009

[13] H-C Lai and J-C Liu ldquoDuality for nondifferentiable minimaxprogramming in complex spacesrdquo Nonlinear Analysis TheoryMethods and Applications vol 71 no 12 pp e224ndashe233 2009

[14] I Ahmad Z Husain and S Sharma ldquoSecond-order dualityin nondifferentiable minmax programming involving type-Ifunctionsrdquo Journal of Computational and Applied Mathematicsvol 215 no 1 pp 91ndash102 2008

[15] I Ahmad Z Husain and S Sharma ldquoHigher-order duality innondifferentiableminimax programmingwith generalized typeI functionsrdquo Journal of Optimization Theory and Applicationsvol 141 no 1 pp 1ndash12 2009

[16] BMond andBD Craven ldquoA class of nondifferentiable complexprogramming problemsrdquo Journal of Mathematics and Statisticsvol 6 pp 581ndash591 1975

[17] S K Mishra and N G Rueda ldquoSymmetric duality for math-ematical programming in complex spaces with F-convexityrdquoJournal of Mathematical Analysis and Applications vol 284 no1 pp 250ndash265 2003

[18] H C Lai and J C Liu ldquoMinimax programming in complexspaces-necessary and sufficient optimality conditionsrdquo in Pro-ceedings of the International Symposium on Nonlinear Analysisand Convex Analysis 2007 Research Institute forMathematicalSciences Kyoto University Japan Rims Kokuroku 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Journal of Optimization

and definitions in complex spaces In Section 3 we establishsufficient optimality conditions under generalized convexityassumptions Weak strong and strict converse duality the-orems related to nondifferentiable minimax programmingproblems in complex spaces for two types of dual models areestablished in Sections 4 and 5 followed by the conclusion inSection 6

2 Notations and Preliminaries

We use the following notations that appear in most works onmathematical programming in complex space

119862119899(119877119899) = 119899-dimensional vector space of complex

(real) numbers119862119898times119899(119877119898times119899)= the set of119898times119899 complex (real)matrices

119877119899+

= 119909 isin 119877119899 119909119895

ge 0 119895 = 1 2 119899 = thenonnegative orthant of 119877119899

119860119867 = 119860119879 = the conjugate transpose of 119860 = [119886

119894119895]

⟨119911 119906⟩ = 119906119867119911 = the inner product of 119906 119911 in 119862119899

Now we recall some definitions related to mathematicalprogramming in complex space that are used in the sequel ofthe paper

Definition 1 (see [5]) A subset 119878 sube 119862119899 is polyhedral cone if

there is 119896 isin 119873 and 119860 isin 119862119899times119896 such that 119878 = 119860119877119896+

= 119860119909 | 119909 isin

119877119896+ that is 119878 is generated by a finite number of vectors (the

columns of 119860)Equivalently 119878 sube 119862

119899 is said to be a polyhedral cone ifit is the intersection of a finite number of closed half-spaceshaving the origin on the boundary that is there is a naturalnumber 119901 and 119901-points 119906

1 1199062 119906

119901such that

119878 =

119901

⋂119896=1

119863(119906119896) = 119911 isin 119862

119899

| Re ⟨119911 119906119896⟩ ge 0 119896 = 1 2 119901

(1)

where 119863(119906119896) 119896 = 1 2 119901 are closed half-spaces involving

the point 119906119896

Definition 2 (see [5]) If 0 = 119878 sub 119862119899 then 119878lowast = 119910 isin 119862119899 |

for all 119911 isin 119878 rArr Re(119910119867119911) ge 0 constitute the dual (polar) of119878

IfΘ 119862119899 rarr 119862 is analytic in a neighbourhood of 119911

0isin 119862119899

thennabla119911Θ(1199110) = [120597Θ(119911

0)120597119911119894] 119894 = 1 2 119899 is the gradient of

functionΘ at 1199110 Similarly if the complex functionΘ(1199081 1199082)

is analytic in 2119899 variables (1199081 1199082) and (1199110 1199110) isin 1198622119899 we

define the gradients by

nabla119911Θ(1199110 1199110) = [

120597Θ (1199110 1199110)

1205971199081119895

] 119895 = 1 2 119899

nabla119911Θ(1199110 1199110) = [

120597Θ (1199110 1199110)

1205971199082119895

] 119895 = 1 2 119899

(2)

In this paper we consider the following complex program-ming problem

min120577isin119883

sup120578isin119884

Re [119891 (120577 120578) + (119911119867119860119911)12

]

subject to 120577 isin 119883 = 120577 = (119911 119911) isin 1198622119899 | minusℎ (120577) isin 119878

(P)

where 119884 = 120578 = (119908119908) | 119908 isin 119862119898 is a compact subset in1198622119898 119860 isin 119862119899times119899 is a positive semidefinite Hermitian matrix 119878is a polyhedral cone in 119862119901 119891(sdot sdot) is continuous and for each120578 isin 119884 119891(sdot 120578) 1198622119899 rarr 119862 and ℎ(sdot) 1198622119899 rarr 119862119901 are analytic in119876 = (119911 119911) | 119911 isin 119862119899 sub 1198622119899 where119876 is a linear manifold overa real field In order to have a convex real part for a nonlinearanalytic function the complex functions need to be definedon the linear manifold over 119877 that is 119876 = 120577 = (119911 119911) isin 1198622119899 |

119911 isin 119862119899

Special Cases (i) If problem (P) is a real programming prob-lem with two variables nondifferentiable minimax problemit may be expressed as

min sup119910isin119884

119891 (119909 119910) + (119909119879119861119909)12

st 119892 (119909) le 0 119909 isin 119877119899

(3)

where 119884 is compact subset of 119877119897 119891(sdot sdot) 119877119899 times 119877119897 rarr 119877 and119892(sdot) 119877119899 rarr 119877119898 are continuously differentiable functions at119909 isin 119877119899 and 119861 is a positive semidefinite symmetric matrixThis problem was studied by Ahmad et al [14 15]

(ii) If 119884 vanishes in (P) then problem (P) reduces to theproblem considered by Mond and Craven [16] that is

min Re [119891 (120577) + (119911119867119860119911)

12

]

st 120577 isin 119883 = 120577 isin 1198622119899 | minusℎ (120577) isin 119878 120577 = (119911 119911) 119911 isin 119862119899

(P1)

(iii) If 119860 = 0 then (P) becomes a differentiable complexminimax programming problem studied by Datta and Bhatia[3] that is

min120577isin119883

sup120578isin119884

Re119891 (120577 120578)

st 120577 isin 119883 = 120577 isin 1198622119899 | minusℎ (120577) isin 119878 (P0)

Definition 3 A functional 119865 119862119899 times 119862119899 times 119862119899 rarr 119877 is said tobe sublinear in its third variable if for any 119911

1 1199112

isin 119862119899 thefollowing conditions are satisfied

(i) 119865(1199111 1199112 1199061+ 1199062) le 119865(119911

1 1199112 1199061) + 119865(119911

1 1199112 1199062)

(ii) 119865(1199111 1199112 120572119906) = 120572119865(119911

1 1199112 119906)

for any 120572 ge 0 in 119877+and 119906

1 1199062 119906 isin 119862119899 From (ii) it is clear

that 119865(1199111 1199112 0) = 0

Let 119865 119862119899 times 119862119899 times 119862119899 rarr 119877 be sublinear on the thirdvariable 120579 119862119899 times 119862119899 rarr 119877

+with 120579(119911

1 1199112) = 0 if 119911

1= 1199112and

120572 119862119899 times 119862119899times rarr 119877+ 0 Let 119891 and ℎ be analytic functions

and 120588 let be a real number Now we introduce the followingdefinitions which are extensions of the definitions given byLai et al [9] and Mishra and Rueda [17]

Journal of Optimization 3

Definition 4 The real part Re[119891] of analytic function 119891 119876 sub

1198622119899 rarr 119862 is said to be (119865 120572 120588 120579)-convex (strict (119865 120572 120588 120579)-convex) with respect to 119877

+on the manifold 119876 = 120577 = (119911 119911) |

119911 isin 119862119899 sub 1198622119899 if for any 120577 = (119911 119911) 1205770= (1199110 1199110) isin 119876 one has

Re [119891 (119911 119911) minus 119891 (1199110 1199110)]

ge (gt) 119865 (119911 1199110 120572 (119911 119911

0)

times (nabla119911119891 (1199110 1199110) + nabla119911119891 (1199110 1199110))) + 120588120579

2

(119911 1199110)

(4)

Definition 5 The real part Re[119891] of analytic function 119891

119876 sub 1198622119899 rarr 119862 is said to be (119865 120572 120588 120579)-quasiconvex (strict(119865 120572 120588 120579)-quasiconvex) with respect to 119877

+on the manifold

119876 = 120577 = (119911 119911) | 119911 isin 119862119899 sub 1198622119899 if for any 120577 = (119911 119911)1205770= (1199110 1199110) isin 119876 one has

Re [119891 (119911 119911) minus 119891 (1199110 1199110)]

le (lt) 0 997904rArr 119865(119911 1199110 120572 (119911 119911

0)

times (nabla119911119891 (1199110 1199110) + nabla119911119891 (1199110 1199110)))

le minus1205881205792

(119911 1199110)

(5)

Definition 6 The real part Re[119891] of analytic function 119891

119876 sub 1198622119899 rarr 119862 is said to be (119865 120572 120588 120579)-pseudoconvex (strict(119865 120572 120588 120579)-pseudoconvex) with respect to119877

+on themanifold

119876 = 120577 = (119911 119911) | 119911 isin 119862119899 sub 1198622119899 if for any 120577 = (119911 119911)1205770= (1199110 1199110) isin 119876 one has

119865 (119911 1199110 120572 (119911 119911

0) (nabla119911119891 (1199110 1199110) + nabla119911119891 (1199110 1199110)))

ge minus1205881205792

(119911 1199110) 997904rArr Re [119891 (119911 119911) minus 119891 (119911

0 1199110)]

ge (gt) 0

(6)

Definition 7 The mapping ℎ 1198622119899 rarr 119862119901 is said to be(119865 120572 120588 120579)-convex (strict (119865 120572 120588 120579)-convex) with respect tothe polyhedral cone 119878 sub 119862

119901 on the manifold 119876 if for any120583 isin 119878 and 120577 = (119911 119911) 120577

0= (1199110 1199110) isin 119876 one has

Re ⟨120583 ℎ (120577) minus ℎ (1205770)⟩

ge (gt) 119865 (119911 1199110 120572 (119911 119911

0)

times (120583119879

nabla119911ℎ (1205770) + 120583119867

nabla119911ℎ (1205770))) + 120588120579

2

(119911 1199110)

(7)

Definition 8 The mapping ℎ 1198622119899 rarr 119862119901 is said to be(119865 120572 120588 120579)-quasiconvex (strict (119865 120572 120588 120579)-quasiconvex) with

respect to the polyhedral cone 119878 sub 119862119901 on the manifold 119876 if

for any 120583 isin 119878 and 120577 = (119911 119911) 1205770= (1199110 1199110) isin 119876 one has

Re ⟨120583 ℎ (120577) minus ℎ (1205770)⟩

le (lt) 0 997904rArr 119865(119911 1199110 120572 (119911 119911

0)

times (120583119879

nabla119911ℎ (1205770) + 120583119867

nabla119911ℎ (1205770)))

le minus1205881205792

(119911 1199110)

(8)

Definition 9 The mapping ℎ 1198622119899 rarr 119862119901 is said to be(119865 120572 120588 120579)-pseudoconvex (strict (119865 120572 120588 120579)-pseudoconvex)with respect to the polyhedral cone 119878 sub 119862

119901 on the manifold119876 if for any 120583 isin 119878 and 120577 = (119911 119911) 120577

0= (1199110 1199110) isin 119876 one has

119865(119911 1199110 120572 (119911 119911

0) (120583119879

nabla119911ℎ (1205770) + 120583119867

nabla119911ℎ (1205770)))

ge 1205881205792

(119911 1199110) 997904rArr Re ⟨120583 ℎ (120577) minus ℎ (120577

0)⟩

ge (gt) 0

(9)

Remark 10 In the proofs of theorems sometimes it may bemore convenient to use certain alternative but equivalentforms of the above definitions Consider the following exam-ple

The real part Re[119891] of analytic function119891 119876 sub 1198622119899 rarr 119862

is said to be (119865 120572 120588 120579)-pseudoconvex with respect to 119877+on

the manifold 119876 = 120577 = (119911 119911) | 119911 isin 119862119899 sub 1198622119899 if for any1205770= (1199110 1199110) isin 119876 one has

Re [119891 (119911 119911) minus 119891 (1199110 1199110)]

lt 0 997904rArr 119865(119911 1199110 120572 (119911 119911

0)

times (nabla119911119891 (1199110 1199110) + nabla119911119891 (1199110 1199110)))

lt minus1205881205792

(119911 1199110)

(10)

Remark 11 If we take 120572(119911 1199110) = 1 then the above definitions

reduce to that given by Lai et al [9] In addition if we take120588 = 0 then we obtain the definitions given by Mishra andRueda [17]

Let 119860 isin 119862119899times119899 and 119911 119906 isin 119862119899 then Schwarz inequality can

be written as

Re (119911119867119860119906) le (119911119867

119860119911)12

(119906119867

119860119906)12

(11)

The equality holds if 119860119911 = 120582119860119906 or 119911 = 120582119906 for 120582 ge 0

Definition 12 (see [12]) The problem (P) is said to satisfy theconstraint qualification at a point 120577

0= (1199110 1199110) if for any

nonzero 120583 isin 119878lowast sub 119862119901

Re ⟨ℎ1015840120577(1205770) (120577 minus 120577

0) 120583⟩ = 0 for 120577 = 120577

0 (12)

In the next section we recall some notations and discussnecessary and sufficient optimality conditions for problem(P) on the basis of Lai and Liu [18] and Lai and Huang [12]

4 Journal of Optimization

3 Necessary and Sufficient Conditions

Let 119891(120577 sdot) 120577 = (119911 119911) isin 1198622119899 be a continuous function definedon119884 where119884 sub 1198622119898 is a specified compact subset in problem(P) Then the supremum sup]isin119884 Re119891(120577 ]) will be attained toits maximum in 119884 and the set

119884 (120577) = 120578 isin 119884 | Re119891 (120577 120578) = sup]isin119884

Re119891 (120577 ]) (13)

is then also a compact set in 1198622119898 In particular if 120577 = 1205770

=

(1199110 1199110) is an optimal solution of problem (P) there exist a

positive integer 119896 and finite points 120578119894isin 119884(120577

0) 120582119894gt 0 119894 =

1 2 119896 withsum119896

119894=1120582119894= 1 such that the Lagrangian function

120601 (120577) =

119896

sum119894=1

120582119894119891 (120577 120578

119894) + ⟨ℎ (120577) 120583⟩ (120583 = 0 in 119878

lowast

) (14)

satisfies the Kuhn-Tucker type condition at 1205770 That is

(

119896

sum119894=1

1205821198941198911015840

120577(1205770 120578119894) + ⟨ℎ

1015840

120577(1205770) 120583⟩) (120577 minus 120577

0) = 0 (15)

Re ⟨ℎ (1205770) 120583⟩ = 0 (16)

Equivalent form of expression (15) at 120577 = 1205770isin 119876 is

119896

sum119894=1

120582119894[nabla119911119891 (1205770 120578119894) + nabla119911119891 (1205770 120578119894)]

+ (120583119879

nabla119911ℎ (1205770) + 120583119867

nabla119911ℎ (1205770)) = 0

(17)

For the integer 119896 corresponding a vector 120578 equiv (1205781 1205782 120578

119896) isin

119884(1205770)119896 and 120582

119894gt 0 119894 = 1 2 119896 withsum

119896

119894=1120582119894= 1 we define a

set as follows

119885120578(1205770) =

120577 isin 1198622119899

| minusℎ1015840

120577(1205770) 120577 isin 119878 (minusℎ (120577

0))

120577 = (119911 119911) isin 119876

Re[119896

sum119894=1

1205821198941198911015840

120577(1205770 120578119894) 120577 + ⟨119860119911 119911⟩

12

] lt 0

(18)

where the set 119878(1199040) is the intersection of closed half-spaces

having the point 1199040isin 119878 on their boundaries

Theorem 13 (necessary optimality conditions) Let 1205770

=

(1199110 1199110) isin 119876 be an optimal solution to (P) Suppose that the

constraint qualification is satisfied for (P) at 1205770and 119911119867

01198601199110

=

⟨1198601199110 1199110⟩ gt 0 Then there exist 0 = 120583 isin 119878lowast sub 119862119901 119906 isin 119862119899 and a

positive integer 119896 with the following properties

(i) 120578119894isin 119884(1205770) 119894 = 1 2 119896

(ii) 120582119894gt 0 119894 = 1 2 119896 sum119896

119894=1120582119894= 1

such that sum119896

119894=1120582119894119891(120577 120578

119894) + ⟨ℎ(120577) 120583⟩ + ⟨119860119911 119911⟩

12 satisfies thefollowing conditions

119896

sum119894=1

120582119894nabla119911119891 (1205770 120578119894) + nabla119911119891 (1205770 120578119894) + 119860119906

+ (120583119879

nabla119911ℎ (1205770) + 120583119867

nabla119911ℎ (1205770)) = 0

(19)

Re ⟨ℎ (1205770) 120583⟩ = 0 (20)

119906119867

119860119906 le 1 (21)

(119911119867

01198601199110)12

= Re (1199111198670119860119906) (22)

Theorem 14 (sufficient optimality conditions) Let 1205770

=

(1199110 1199110) isin 119876 be a feasible solution to (P) Suppose that there

exists a positive integer 119896 120582119894gt 0 120578119894isin 119884(1205770) 119894 = 1 2 119896with

sum119896

119894=1120582119894= 1 and 0 = 120583 isin 119878lowast sub 119862119901 satisfying conditions (19)ndash

(22) Further if Re[sum119896119894=1

120582119894[119891(120577 120578

119894) + 119911119867119860119906]] is (119865 120572

1 1205881 120579)-

convex with respect to 119877+on 119876 ℎ(120577) is (119865 120572

2 1205882 120579)-convex on

119876with respect to the polyhedral cone 119878 sub 119862119901 and12058811205721(119911 1199110)+

12058821205722(119911 1199110) ge 0 then 120577

0= (1199110 1199110) is an optimal solution to

(P)

Proof We prove this theorem by contradiction Suppose thatthere is a feasible solution 120577 isin 119876 such that

sup120578isin119884

Re [119891 (120577 120578) + (119911119867

119860119911)12

]

lt sup120578isin119884

Re [119891 (1205770 120578) + (119911

119867

01198601199110)12

]

(23)

Since 120578119894isin 119884(1205770) 119894 = 1 2 119896 we have

sup120578isin119884

Re [119891 (1205770 120578) + (119911

119867

01198601199110)12

]

= Re [119891 (1205770 120578119894) + (119911

119867

01198601199110)12

]

for 119894 = 1 2 119896

Re [119891 (120577 120578119894) + (119911

119867

119860119911)12

]

lt sup120578isin119884

Re [119891 (120577 120578) + (119911119867

119860119911)12

]

for 119894 = 1 2 119896

(24)

Thus from the above three inequalities we obtain

Re [119891 (120577 120578119894) + (119911

119867

119860119911)12

]

lt Re [119891 (1205770 120578119894) + (119911

119867

01198601199110)12

]

for 119894 = 1 2 119896

(25)

Journal of Optimization 5

Using (21) and generalized Schwarz inequality we get

Re (119911119867119860119906) le (119911119867

119860119911)12

(119906119867

119860119906)12

le (119911119867

119860119911)12

= Re [(119911119867119860119911)12

]

(26)

and inequality (22) yields

Re (1199111198670119860119906) = Re [(119911119867

01198601199110)12

] (27)

Using (26) and (27) in (25) we have

Re [119891 (120577 120578119894) + 119911119867

119860119906] lt Re [119891 (1205770 120578119894) + 119911119867

0119860119906]

for 119894 = 1 2 119896

(28)

Since 120582119894gt 0 and sum

119896

119894=1120582119894= 1 we have

119903119897Re[119896

sum119894=1

120582119894[119891 (120577 120578

119894) + 119911119867

119860119906] minus

119896

sum119894=1

120582119894[119891 (1205770 120578119894) + 119911119867

0119860119906]]

lt 0

(29)

Since Re[sum119896119894=1

120582119894[119891(120577 120578

119894) + 119911119867119860119906]] is (119865 120572

1 1205881 120579)-convex

with respect to 119877+on 119876 we have

Re[119896

sum119894=1

120582119894[119891 (120577 120578

119894) + 119911119867

119860119906]

minus

119896

sum119894=1

120582119894[119891 (1205770 120578119894) + 119911119867

0119860119906]]

ge 119865[119911 1199110 1205721(119911 1199110)

times(

119896

sum119894=1

120582119894nabla119911119891 (1205770 120578119894) + nabla119911119891 (1205770 120578119894) + 119860119906)]

+ 12058811205792

(119911 1199110)

(30)

From (29) and (30) we conclude that

119865[119911 1199110 1205721(119911 1199110)

times(

119896

sum119894=1

120582119894nabla119911119891(1205770 120578119894) + nabla119911119891 (1205770 120578119894) + 119860119906)]

lt minus12058811205792

(119911 1199110)

(31)

which due to sublinearity of 119865 can be written as

119865[119911 1199110

119896

sum119894=1

120582119894nabla119911119891 (1205770 120578119894) + nabla119911119891 (1205770 120578119894) + 119860119906]

lt minus1205881

1205721(119911 1199110)1205792

(119911 1199110)

(32)

On the other hand from the feasibility of 120577 to (P) we haveminusℎ(120577) isin 119878 or Re⟨ℎ(120577) 120583⟩ le 0 for 120583 isin 119878lowast which along with(20) yields

Re ⟨ℎ (120577) 120583⟩ le 0 = Re ⟨ℎ (1205770) 120583⟩ (33)

Since ℎ(120577) is (119865 1205722 1205882 120579)-convex on 119876 with respect to the

polyhedral cone 119878 sub 119862119901 we have

Re ⟨ℎ (120577) 120583⟩ minus Re ⟨ℎ (1205770) 120583⟩

ge 119865 [119911 1199110 1205722(119911 1199110) (120583119879

nabla119911ℎ (1205770) + 120583119867

nabla119911ℎ (1205770))]

+ 12058821205792

(119911 1199110)

(34)

From (33) and (34) it follows that

119865 [119911 1199110 1205722(119911 1199110) (120583119879

nabla119911ℎ (1205770) + 120583119867

nabla119911ℎ (1205770))]

le minus12058821205792

(119911 1199110)

(35)

which due to sublinearity of 119865 can be written as

119865 [119911 1199110 120583119879

nabla119911ℎ (1205770) + 120583119867

nabla119911ℎ (1205770)] le minus

1205882

1205722(119911 1199110)1205792

(119911 1199110)

(36)

On adding (32) and (36) and using sublinearity of 119865 we get

119865[119911 1199110

119896

sum119894=1

120582119894nabla119911119891 (1205770 120578119894) + nabla119911119891 (1205770 120578119894) + 119860119906

+120583119879

nabla119911ℎ (1205770) + 120583119867

nabla119911ℎ (1205770) ]

lt minus(1205881

1205721(119911 1199110)+

1205882

1205722(119911 1199110)) 1205792

(119911 1199110)

(37)

The above inequality together with the assumption12058811205721(119911 1199110) + 12058821205722(119911 1199110) ge 0 gives

119865[119911 1199110

119896

sum119894=1

120582119894nabla119911119891 (1205770 120578119894) + nabla119911119891 (1205770 120578119894) + 119860119906

+ 120583119879

nabla119911ℎ (1205770) + 120583119867

nabla119911ℎ (1205770) ] lt 0

(38)

which contradicts (19) hence the theorem

6 Journal of Optimization

Theorem 15 (sufficient optimality conditions) Let 1205770

=

(1199110 1199110) isin 119876 be a feasible solution to (P) Suppose that there

exists a positive integer 119896 120582119894gt 0 120578119894isin 119884(1205770) 119894 = 1 2 119896with

sum119896

119894=1120582119894= 1 and 0 = 120583 isin 119878lowast sub 119862119901 satisfying conditions (19)ndash

(22) Further if Re[sum119896119894=1

120582119894[119891(120577 120578

119894) + 119911119867119860119906]] is (119865 120572

1 1205881 120579)-

pseudoconvex with respect to 119877+on 119876 ℎ(120577) is (119865 120572

2 1205882 120579)-

quasiconvex on 119876 with respect to the polyhedral cone 119878 sub 119862119901and 120588

11205721(119911 1199110) + 12058821205722(119911 1199110) ge 0 then 120577

0= (1199110 1199110) is an

optimal solution to (P)

Proof Proceeding as in Theorem 14 we have

Re[119896

sum119894=1

120582119894[119891 (120577 120578

119894) + 119911119867

119860119906] minus

119896

sum119894=1

120582119894[119891 (1205770 120578119894) + 119911119867

0119860119906]]

lt 0

(39)

which by (119865 1205721 1205881 120579)-pseudoconvexity of

Re[sum119896119894=1

120582119894[119891(120577 120578

119894) + 119911119867119860119906]] with respect to 119877

+on 119876

yields

119865[119911 1199110 1205721(119911 1199110)

119896

sum119894=1

120582119894nabla119911119891 (1205770 120578119894) + nabla119911119891 (1205770 120578119894) + 119860119906]

lt minus12058811205792 (119911 119911

0)

(40)

Using the sublinearity of 119865 the above inequality can bewritten as

119865[119911 1199110

119896

sum119894=1

120582119894nabla119911119891 (1205770 120578119894) + nabla119911119891 (1205770 120578119894) + 119860119906]

lt minus1205881

1205721(119911 1199110)1205792 (119911 119911

0)

(41)

On the other hand from the feasibility of 120577 to (P) we haveminusℎ(120577) isin 119878 or Re⟨ℎ(120577) 120583⟩ le 0 for 120583 isin 119878lowast which along with(20) yields

Re ⟨ℎ (120577) 120583⟩ le 0 = Re ⟨ℎ (1205770) 120583⟩ (42)

Since ℎ(120577) is (119865 1205722 1205882 120579)-quasiconvex on 119876 with respect to

the polyhedral cone 119878 sub 119862119901 the above inequality yields

119865 [119911 1199110 1205722(119911 1199110) (120583119879

nabla119911ℎ (1205770) + 120583119867

nabla119911ℎ (1205770))]

le minus12058821205792 (119911 119911

0)

(43)

which due to sublinearity of 119865 can be written as

119865 [119911 1199110 120583119879

nabla119911ℎ (1205770) + 120583

119867

nabla119911ℎ (1205770)] le minus

1205882

1205722(119911 1199110)1205792

(119911 1199110)

(44)

On adding (41) and (44) and using sublinearity of 119865 we get

119865[119911 1199110

119896

sum119894=1

120582119894nabla119911119891 (1205770 120578119894) + nabla119911119891 (1205770 120578119894) + 119860119906

+ 120583119879

nabla119911ℎ (1205770) + 120583119867

nabla119911ℎ (1205770) ]

lt minus(1205881

1205721(119911 1199110)+

1205882

1205722(119911 1199110)) 1205792

(119911 1199110)

(45)

The above inequality together with the assumption12058811205721(119911 1199110) + 12058821205722(119911 1199110) ge 0 gives

119865[119911 1199110

119896

sum119894=1

120582119894nabla119911119891 (1205770 120578119894) + nabla119911119891 (1205770 120578119894) + 119860119906

+120583119879

nabla119911ℎ (1205770) + 120583119867

nabla119911ℎ (1205770) ] lt 0

(46)

which contradicts (19) hence the theorem

Theorem 16 (sufficient optimality conditions) Let 1205770

=

(1199110 1199110) isin 119876 be a feasible solution to (P) Suppose that there

exists a positive integer 119896 120582119894gt 0 120578119894isin 119884(1205770) 119894 = 1 2 119896with

sum119896

119894=1120582119894= 1 and 0 = 120583 isin 119878lowast sub 119862119901 satisfying conditions (19)ndash

(22) Further if Re[sum119896119894=1

120582119894[119891(120577 120578

119894) + 119911119867119860119906]] is (119865 120572

1 1205881 120579)-

quasiconvex with respect to 119877+on119876 ℎ(120577) is strict (119865 120572

2 1205882 120579)-

pseudoconvex on119876 with respect to the polyhedral cone 119878 sub 119862119901

and 12058811205721(119911 1199110) + 12058821205722(119911 1199110) ge 0 then 120577

0= (1199110 1199110) is an

optimal solution to (P)

Proof The proof follows on the similar lines of Theorem 15

4 Parametric Duality

We adopt the following notations in order to simplify theformulation of dual

119870 (120585) = (119896 120578) isin 119873 times 119877119896

+times 1198622119898119896

|

= (1205821 1205822 120582

119896) with

119896

sum119894=1

120582119894= 1

120578 = (1205781 1205782 120578

119896)

with 120578119894isin 119884 (120585) 119894 = 1 2 119896

(47)

for 120585 = (119906 119906) isin 119876 sub 1198622119899Now we formulate a parametric dual problem (D1) with

respect to the complex minimax programming problem (P)as follows

max(119896120582120578)isin119870(120585)

sup(120585120583119908119905)isin119883(119896120582120578)

119905(D1)

Journal of Optimization 7

where119883(119896 120578) denotes the set of all (120585 120583 119908 119905) isin 1198622119899 times119862119901 times

119862119899 times 119877 to satisfy the following conditions119896

sum119894=1

120582119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]

+ 120583119879 nabla119911ℎ (120585) + 120583119867nabla

119911ℎ (120585) = 0

(48)

119896

sum119894=1

120582119894Re119891 (120585 120578

119894) + (119906

119867

119860119906)12

minus 119905 ge 0 (49)

Re ⟨ℎ (120585) 120583⟩ ge 0 (50)

119908119867

119860119908 le 1 (51)

(119906119867

119860119906)12

= Re (119906119867119860119908) (52)

0 = 120583 isin 119878lowast

(53)

If for a triplet (119896 120578) isin 119870(120585) the set 119883(119896 120578) = 0then we define the supremum over 119883(119896 120578) to be minusinfin fornonexecption in the formulation of (D1)

Theorem 17 (weak duality) Let 120577 = (119911 119911) and(119896 120578 120585 120583 119908 119905) be feasible solutions to (P) and (D1)respectively Further if Re[sum119896

119894=1120582119894[119891(120577 120578

119894) + 119911119867119860119908]] is

(119865 1205721 1205881 120579)-pseudoconvex with respect to 119877

+on 119876 ℎ(120577) is

(119865 1205722 1205882 120579)-quasiconvex on 119876 with respect to the polyhedral

cone 119878 sub 119862119901 and 12058811205721(119911 119906) + 120588

21205722(119911 119906) ge 0 Then

sup120578isin119884

Re [119891 (120577 120578) + (119911119867

119860119911)12

] ge 119905 (54)

Proof Suppose on the contrary that

sup120578isin119884

Re [119891 (120577 120578) + (119911119867

119860119911)12

] lt 119905 (55)

By compactness of 119884(120585) sub 119884 in 119862119901 120585 isin (119906 119906) isin 119876 thereexist an integer 119896 gt 0 and finite points 120578

119894isin 119884(120585) 120582

119894gt 0

119894 = 1 2 119896 with sum119896

119894=1120582119894= 1 such that (49) holds From

(49) and (55) we have

Re[119896

sum119894=1

120582119894[119891 (120577 120578

119894) + (119911

119867

119860119911)12

]] lt

119896

sum119894=1

120582119894119905

le

119896

sum119894=1

120582119894Re [119891 (120585 120578

119894) + (119906

119867

119860119906)12

]

(56)

From (51) and the generalized Schwarz inequality we have

Re (119911119867119860119908) le (119911119867

119860119911)12

(119908119867

119860119908)12

le (119911119867

119860119911)12

(57)

Using (52) and (57) in (56) we get

Re[119896

sum119894=1

120582119894[119891 (120577 120578

119894) + 119911119867

119860119908]]

lt Re[119896

sum119894=1

120582119894[119891 (120585 120578

119894) + 119906119867

119860119908]]

(58)

Since Re[sum119896119894=1

120582119894[119891(120577 120578

119894) + 119911119867119860119908]] is (119865 120572

1 1205881 120579)-

pseudoconvex with respect to 119877+on 119876 the above inequality

implies that

119865[119911 119906 1205721(119911 119906)

119896

sum119894=1

120582119894nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]

lt minus12058811205792

(119911 119906)

(59)

which due to sublinearity of 119865 can be written as

119865[119911 119906

119896

sum119894=1

120582119894nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]

lt minus1205881

1205721(119911 119906)

1205792

(119911 119906)

(60)

By the feasibility of 120577 = (119911 119911) to (P) we have minusℎ(120577) isin 119878 orRe⟨ℎ(120577) 120583⟩ le 0 for 120583 isin 119878lowast which along with (50) yields

Re ⟨ℎ (120577) 120583⟩ le 0 le Re ⟨ℎ (120585) 120583⟩ (61)

The above inequality together with the (119865 1205722 1205882 120579)-

quasiconvexity of ℎ(120577) on 119876 with respect to the polyhedralcone 119878 sub 119862

119901 implies

119865 [119911 119906 1205722(119911 119906) (120583

119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

le minus12058821205792

(119911 119906)

(62)

which due to sublinearity of 119865 can be written as

119865 [119911 119906 120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585)]

le minus1205882

1205722(119911 119906)

1205792

(119911 119906) (63)

On adding (60) and (63) and using sublinearity of 119865 we get

119865[119911 119906

119896

sum119894=1

120582119894nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908

+ 120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585) ]

lt minus(1205881

1205721(119911 119906)

+1205882

1205722(119911 119906)

) 1205792

(119911 119906)

(64)

From the assumption 12058811205721(119911 119906) +120588

21205722(119911 119906) ge 0 the above

inequality yields

119865[119911 119906

119896

sum119894=1

120582119894nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908

+ 120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585) ] lt 0

(65)

which contradicts (48) hence the theorem

8 Journal of Optimization

Theorem 18 (weak duality) Let 120577 = (119911 119911) and(119896 120578 120585 120583 119908 119905) be feasible solutions to (P) and (D1)respectively Further if Re[sum119896

119894=1120582119894[119891(120577 120578

119894) + 119911

119867

119860119908]] is(119865 1205721 1205881 120579)-quasiconvex with respect to 119877

+on 119876 ℎ(120577) is

(119865 1205722 1205882 120579)-pseudoconvex on 119876 with respect to the polyhedral

cone 119878 sub 119862119901 and 12058811205721(119911 119906) + 120588

21205722(119911 119906) ge 0 then

sup120578isin119884

Re [119891 (120577 120578) + (119911119867

119860119911)12

] ge 119905 (66)

Proof Theproof follows the same lines as inTheorem 17

Theorem 19 (strong duality) Let 1205770= (1199110 1199110) be an optimal

solution to the problem (P) at which a constraint qualificationis satisfied Then there exist (119896 120578) isin 119870(120577

0) and (120577

0 120583 119908 119905) isin

119883(119896 120578) such that (119896 120578 1205770 120583 119908 119905) is a feasible solution to

the dual problem (D1) If the hypotheses of Theorem 17 or 18are satisfied then (119896 120578 120577

0 120583 119908 119905) is optimal to (D1) and

the two problems (P) and (D1) have the same optimal values

Proof Theproof follows along the lines ofTheorem6 (Lai andLiu [13])

Theorem 20 (strict converse duality) Let 120577 and(

120582 120578 120585 120583 119908 ) be optimal solutions to (P) and (D1)

respectively and assume that the assumptions of Theorem 19are satisfied Further assume that the following conditions aresatisfied

(i) Resum119896

119894=1119894[119891(120577 120578

119894) + 119867119860119908] is strict (119865 120572

1 1205881 120579)-

pseudoconvex with respect to 119877+on 119876 and ℎ(120577) is

(119865 1205722 1205882 120579)-quasiconvex on 119876 with respect to the

polyhedral cone 119878 sub 119862119901(ii) 12058811205721( ) + 120588

21205722( ) ge 0

Then 120577 = 120585 that is 120577 is optimal solution to (D1)

Proof On the contrary suppose that ( ) = 120577 = 120585 = ( )On applyingTheorem 19 we know that

= supisin119884

Re [119891 (120577 120578) + (119867

119860)12

] (67)

From the feasibility of 120577 isin 119876 to (P) 120583 isin 119878lowast and (50) we have

Re ⟨ℎ (120577) 120583⟩ le 0 le Re ⟨ℎ (120585) 120583⟩ (68)

Since ℎ(120577) is (119865 1205722 1205882 120579)-quasiconvex on 119876 with respect to

the polyhedral cone Sin 119862119901 the above inequality yields

119865 [ 1205722( ) (120583

119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

le minus12058821205792

( )

(69)

which by sublinearity of 119865 implies

119865 [ (120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

le minus1205882

1205722( )

1205792

( )

(70)

By (48) and the sublinearity of 119865 we have

119865[

[

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]]

]

+ 119865 [ (120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

ge 119865[

[

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]

+ 120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585)]

]

= 0

(71)

The above inequality together with (70) and 12058811205721( ) +

12058821205722( ) ge 0 gives

119865[

[

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]]

]

ge minus119865 [ (120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

ge1205882

1205722( )

1205792

( )

ge minus1205881

1205721( )

1205792

( )

(72)

That is

119865[

[

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]]

]

ge minus1205881

1205721( )

1205792

( )

(73)

which by sublinearity of 119865 implies

119865[

[

1205721( )

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]]

]

ge minus12058811205792

( )

(74)

Since Resum119896

119894=1119894[119891(120577 120578

119894) + 119867119860119908] is strict (119865 120572

1 1205881 120579)-

pseudoconvex with respect to 119877+on 119876 the above inequality

implies that

Re[

[

119896

sum119894=1

119894[119891 (120577 120578

119894) + 119867

119860119908]]

]

gt Re[

[

119896

sum119894=1

119894[119891 (120585 120578

119894) + 119867

119860119908]]

]

(75)

Journal of Optimization 9

From (51) (52) and the generalized Schwarz inequality wehave

Re (119867119860119908) le (119867

119860)12

Re (119867119860119908) = (119867

119860)12

(76)

which on substituting in (75) and by using (49) we obtain

Re[

[

119896

sum119894=1

119894[119891 (120577 120578

119894) + (

119867

119860)12

]]

]

gt Re[

[

119896

sum119894=1

119894[119891 (120585 120578

119894) + (

119867

119860)12

]]

]

ge

119896

sum119894=1

119894

(77)

Consequently there exist certain 1198940which satisfy

Re [119891 (120577 1205781198940) + (

119867

119860)12

] gt (78)

Hence

supisin119884

Re [119891 (120577 120578) + (119867

119860)12

]

ge Re [119891 (120577 1205781198940) + (

119867

119860)12

] gt

(79)

which contradicts (67) hence the theorem

5 Parameter Free Duality

Making use of the optimality conditions we show thatthe following formation is a dual (D2) to the complexprogramming problem (P)

max(119896120582120578)isin119870(120585)

sup(120585120583119908)isin119883(119896120582120578)

Re [119891 (120585 120578) + (119906119867

119860119906)12

] (D2)

where119883(119896 120578) denotes the set of all (120585 120583 119908) isin 1198622119899times119862119901times119862119899

to satisfy the following conditions

119896

sum119894=1

120582119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]

+ 120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585) = 0

(80)

Re ⟨ℎ (120585) 120583⟩ ge 0 (81)

119908119867

119860119908 le 1 (82)

(119906119867

119860119906)12

= Re (119906119867119860119908) (83)

If for a triplet (119896 120578) isin 119870(120585) the set 119883(119896 120578) = 0then we define the supremum over 119883(119896 120578) to be minusinfin fornonexception in the formulation of (D2)

Nowwe establish appropriate duality theorems and provethat optimal values of (P) and (D2) are equal under theassumption of generalized convexity in order to show that theproblems (P) and (D2) have no duality gap

Theorem21 (weak duality) Let 120577 = (119911 119911) and (119896 120578 120585 120583 119908)

be feasible solutions to (P) and (D2) respectively Furtherif Re[sum119896

119894=1120582119894[119891(120577 120578

119894) + 119911119867119860119908]] is (119865 120572

1 1205881 120579)-pseudoconvex

with respect to 119877+on 119876 ℎ(120577) is (119865 120572

2 1205882 120579)-quasiconvex on 119876

with respect to the polyhedral cone 119878 sub 119862119901 and 12058811205721(z 119906) +

12058821205722(119911 119906) ge 0 then

sup120578isin119884

Re [119891 (120577 120578) + (119911119867

119860119911)12

]

ge sup120578isin119884

Re [119891 (120585 120578) + (119906119867

119860119906)12

]

(84)

Proof On the contrary we suppose that

sup120578isin119884

Re [119891 (120577 120578) + (119911119867

119860119911)12

]

lt sup120578isin119884

Re [119891 (120585 120578) + (119906119867

119860119906)12

]

(85)

Since 120578119894isin 119884(120585) sub 119884 119894 = 1 2 119896 we have

sup120578isin119884

Re [119891 (120585 120578) + (119906119867

119860119906)12

]

= Re [119891 (120585 120578119894) + (119906

119867

119860119906)12

] 119894 = 1 2 119896

(86)

Re [119891 (120577 120578119894) + (119911

119867

119860119911)12

]

le sup120578isin119884

Re [119891 (120577 120578) + (119911119867

119860119911)12

] 119894 = 1 2 119896

(87)

Then the above three inequalities give

Re [119891 (120577 120578119894) + (119911

119867

119860119911)12

]

lt Re [119891 (120585 120578119894) + (119906

119867

119860119906)12

] 119894 = 1 2 119896

(88)

From (82) (83) (88) and the generalized Schwarz inequalitywe have

Re [119891 (120577 120578119894) + (119911

119867

119860119908)]

lt Re [119891 (120585 120578119894) + (119906

119867

119860119908)] 119894 = 1 2 119896

(89)

As 120582119894gt 0 119894 = 1 2 119896 and sum

119896

119894=1120582119894= 1 we have

Re[119896

sum119894=1

120582119894[119891 (120577 120578

119894) + (119911

119867

119860119908)]]

minus Re[119896

sum119894=1

120582119894[119891 (120585 120578

119894) + (119906

119867

119860119908)]] lt 0

(90)

10 Journal of Optimization

Since Re[sum119896119894=1

120582119894[119891(120577 120578

119894) + 119911119867119860119908]] is (119865 120572

1 1205881 120579)-

pseudoconvex with respect to 119877+on 119876 the above inequality

implies that

119865[119911 119906 1205721(119911 119906)

times

119896

sum119894=1

120582119894nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]

lt minus12058811205792

(119911 119906)

(91)

which by sublinearity of 119865 becomes

119865[119911 119906

119896

sum119894=1

120582119894nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]

lt minus1205881

1205721(119911 119906)

1205792

(119911 119906)

(92)

By the feasibility of 120577 = (119911 119911) to (P) 0 = 120583 isin 119878lowast and theinequality (81) we obtain

Re ⟨ℎ (120577) 120583⟩ le 0 le Re ⟨ℎ (120585) 120583⟩ (93)

The above inequality together with the (119865 1205722 1205882 120579)-

quasiconvexity of ℎ(120577) on 119876 with respect to the polyhedralcone 119878 sub 119862

119901 implies

119865 [119911 119906 1205722(119911 119906) (120583

119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

le minus12058821205792

(119911 119906)

(94)

which by sublinearity of 119865 becomes

119865 [119911 119906 120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585)]

le minus1205882

1205722(119911 119906)

1205792

(119911 119906) (95)

On adding (92) and (95) and using the sublinearity of 119865 weget

119865[119911 119906

119896

sum119894=1

120582119894nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908

+120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585) ]

lt minus(1205881

1205721(119911 119906)

+1205882

1205722(119911 119906)

) 1205792

(119911 119906)

(96)

From the assumption 12058811205721(119911 119906) +120588

21205722(119911 119906) ge 0 the above

inequality yields

119865[119911 119906

119896

sum119894=1

120582119894nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908

+ 120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585) ] lt 0

(97)

which contradicts (80) hence the theorem

Theorem 22 (weak duality) Let 120577 = (119911 119911) and(119896 120578 120585 120583 119908) be feasible solutions to (P) and (D2)respectively Further if Re[sum119896

119894=1120582119894[119891(120577 120578

119894) + 119911

119867

119860119908]] is(119865 1205721 1205881 120579)-quasiconvex with respect to 119877

+on 119876 ℎ(120577) is

(119865 1205722 1205882 120579)-pseudoconvex on 119876 with respect to the polyhedral

cone 119878 sub 119862119901 and 12058811205721(119911 119906) + 120588

21205722(119911 119906) ge 0 then

sup120578isin119884

Re [119891 (120577 120578) + (119911119867

119860119911)12

]

ge sup120578isin119884

Re [119891 (120585 120578) + (119906119867

119860119906)12

]

(98)

Proof Theproof follows the same lines as inTheorem 21

Theorem 23 (strong duality) Let 1205770= (1199110 1199110) be an optimal

solution to the problem (P) at which a constraint qualificationis a satisfied Then there exist (119896 120578) isin 119870(120577

0) and (120577

0 120583 119908) isin

119883(119896 120578) such that (119896 120578 1205770 120583 119908) is a feasible solution to the

dual problem (D2) Further if the hypotheses of Theorem 21or Theorem 22 are satisfied then (119896 120578 120577

0 120583 119908) is optimal

to (D2) and the two problems (P) and (D2) have the sameoptimal values

Proof Theproof follows along the lines ofTheorem8 (Lai andLiu [13])

Theorem 24 (strict converse duality) Let 120577 and(

120582 120578 120585 120583 119908) be optimal solutions to (P) and (D2)

respectively and the conditions of Theorem 23 are satisfiedFurther assume that the following conditions are satisfied

(i) Resum119896

119894=1119894[119891(120577 120578

119894) + 119911119867119860119908] is strict (119865 120572

1 1205881 120579)-

pseudoconvex with respect to 119877+on 119876 and ℎ(120577) is

(119865 1205722 1205882 120579)-quasiconvex on 119876 with respect to the

polyhedral cone 119878 sub 119862119901(ii) 12058811205721( ) + 120588

21205722( ) ge 0

Then 120577 = 120585 that is 120585 is an optimal solution to (D2)

Proof On the contrary we assume that ( ) = 120577 = 120585 = ( )On applyingTheorem 23 we know that

supisin119884

Re [119891 (120577 120578) + (119867

119860)12

]

= supisin119884

Re [119891 (120585 120578) + (119867

119860)12

]

(99)

From the feasibility of 120577 isin 119876 to (P) 120583 isin 119878lowast inequality (81)yields

Re ⟨ℎ (120577) 120583⟩ le 0 le Re ⟨ℎ (120585) 120583⟩ (100)

Since ℎ(120577) is (119865 1205722 1205882 120579)-quasiconvex on 119876 with respect to

the polyhedral cone 119878 in 119862119901 the above inequality yields

119865 [ 1205722( ) (120583

119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

le minus12058821205792

( )

(101)

Journal of Optimization 11

which by sublinearity of 119865 implies

119865 [ (120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

le minus1205882

1205722( )

1205792

( )

(102)

By (80) and the sublinearity of 119865 we have

119865[

[

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]]

]

+ 119865 [ (120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

ge 119865[

[

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]

+ 120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585)]

]

= 0

(103)

The above inequality together with (102) and 12058811205721( ) +

12058821205722( ) ge 0 gives

119865[

[

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]]

]

ge minus119865 [ (120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

ge1205882

1205722( )

1205792

( )

ge minus1205881

1205721( )

1205792

( )

(104)

That is

119865[

[

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]]

]

ge minus1205881

1205721( )

1205792 ( )

(105)

which by sublinearity of 119865 implies

119865[

[

1205721( )

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]]

]

ge minus12058811205792 ( )

(106)

Since Resum119896

119894=1119894[119891(120577 120578

119894) + 119911119867119860119908] is strict (119865 120572

1 1205881 120579)-

pseudoconvex with respect to 119877+on 119876 the above inequality

implies that

Re[

[

119896

sum119894=1

119894[119891 (120577 120578

119894) + 119867

119860119908]]

]

gt Re[

[

119896

sum119894=1

119894[119891 (120585 120578

119894) + 119867

119860119908]]

]

(107)

From (82) (83) and the generalized Schwarz inequality weget

Re (119867119860119908) le (119867

119860)12

Re (119867119860119908)12

= (119867

119860)12

(108)

which on substituting in (107) we obtain

Re[

[

119896

sum119894=1

119894[119891 (120577 120578

119894) + (

119867

119860)12

] ]

]

gt Re[

[

119896

sum119894=1

119894[119891 (120585 120578

119894) + (

119867

119860)12

]]

]

(109)

Consequently there exist certain 1198940which satisfy

Re [119891 (120577 1205781198940) + (119867119860)

12

]

gt Re [119891 (120585 1205781198940) + (119867119860)

12

]

(110)

Hence

supisin119884

Re [119891 (120577 120578) + (119867

119860)12

]

ge Re [119891 (120577 1205781198940) + (

119867

119860)12

]

gt Re [119891 (120585 1205781198940) + (

119867

119860)12

]

= supisin119884

Re [119891 (120585 120578) + (119867

119860)12

]

(111)

which contradicts (99) hence the theorem

6 Conclusion

In this paper we introduced generalized (119865 120572 120588 120579)-convexfunctions and established sufficient optimality conditions fora class of nondifferentiable minimax programming problemsin complex space These optimality conditions are thenused to construct two types of dual model and finally wederived weak strong and strict converse duality theorems

12 Journal of Optimization

to show that there is no duality gap between the two dualproblems with respect to the primal problem under somegeneralized convexities of complex functions in the complexprogramming problem As a future task the authors wouldlike to extend these results to second and higher order casesand establish the relations between primal and its second andhigher order dual problems

Acknowledgments

The authors are thankful to the anonymous referees for theirvaluable comments which have improved the presentationof the paper The research of the first author is financiallysupported by the University Grant Commission New DelhiIndia through Grant No F no 41-8012012(SR)

References

[1] N Levinson ldquoLinear programming in complex spacerdquo Journalof Mathematical Analysis and Applications vol 14 no 1 pp 44ndash62 1966

[2] J C Chen and H C Lai ldquoOptimality conditions for minimaxprogramming of analytic functionsrdquoTaiwanese Journal ofMath-ematics vol 8 no 4 pp 673ndash686 2004

[3] N Datta and D Bhatia ldquoDuality for a class of nondifferentiablemathematical programming problems in complex spacerdquo Jour-nal of Mathematical Analysis and Applications vol 101 no 1 pp1ndash11 1984

[4] O P Jain and P C Saxena ldquoA duality theorem for a specialclass of programming problems in complex spacerdquo Journal ofOptimization Theory and Applications vol 16 no 3-4 pp 207ndash220 1975

[5] I M Stancu-Minasian Fractional Programming Theory Meth-ods and Applications vol 409 Kluwer Academic DordrechtGermany 1997

[6] I Ahmad and Z Husain ldquoOptimality conditions and dualityin nondifferentiable minimax fractional programming withgeneralized convexityrdquo Journal of Optimization Theory andApplications vol 129 no 2 pp 255ndash275 2006

[7] A Jayswal I Ahmad and D Kumar ldquoNondifferentiable mini-max fractional programming problem with nonsmooth gener-alized convex functionsrdquoCommunications onAppliedNonlinearAnalysis vol 18 no 4 pp 57ndash75 2011

[8] A Jayswal and D Kumar ldquoOn nondifferentiable minimaxfractional programming involving generalized (C 120572 120588 d)-convexityrdquo Communications on Applied Nonlinear Analysis vol18 no 1 pp 61ndash77 2011

[9] H-C Lai J-C Lee and S-C Ho ldquoParametric duality on mini-max programming involving generalized convexity in complexspacerdquo Journal of Mathematical Analysis and Applications vol323 no 2 pp 1104ndash1115 2006

[10] K Swarup and I C Sharma ldquoProgramming with linear frac-tional functionals in complex spacerdquo Cahiers Center drsquoEtudesRecherche Operationelle vol 12 pp 103ndash109 1970

[11] A Charnes and W W Cooper ldquoProgramming with linearfractional functionalsrdquo Naval Research Logistics Quarterly vol9 no 3-4 pp 181ndash186 1962

[12] H-C Lai and T-Y Huang ldquoOptimality conditions for non-differentiable minimax fractional programming with complexvariablesrdquo Journal of Mathematical Analysis and Applicationsvol 359 no 1 pp 229ndash239 2009

[13] H-C Lai and J-C Liu ldquoDuality for nondifferentiable minimaxprogramming in complex spacesrdquo Nonlinear Analysis TheoryMethods and Applications vol 71 no 12 pp e224ndashe233 2009

[14] I Ahmad Z Husain and S Sharma ldquoSecond-order dualityin nondifferentiable minmax programming involving type-Ifunctionsrdquo Journal of Computational and Applied Mathematicsvol 215 no 1 pp 91ndash102 2008

[15] I Ahmad Z Husain and S Sharma ldquoHigher-order duality innondifferentiableminimax programmingwith generalized typeI functionsrdquo Journal of Optimization Theory and Applicationsvol 141 no 1 pp 1ndash12 2009

[16] BMond andBD Craven ldquoA class of nondifferentiable complexprogramming problemsrdquo Journal of Mathematics and Statisticsvol 6 pp 581ndash591 1975

[17] S K Mishra and N G Rueda ldquoSymmetric duality for math-ematical programming in complex spaces with F-convexityrdquoJournal of Mathematical Analysis and Applications vol 284 no1 pp 250ndash265 2003

[18] H C Lai and J C Liu ldquoMinimax programming in complexspaces-necessary and sufficient optimality conditionsrdquo in Pro-ceedings of the International Symposium on Nonlinear Analysisand Convex Analysis 2007 Research Institute forMathematicalSciences Kyoto University Japan Rims Kokuroku 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Optimization 3

Definition 4 The real part Re[119891] of analytic function 119891 119876 sub

1198622119899 rarr 119862 is said to be (119865 120572 120588 120579)-convex (strict (119865 120572 120588 120579)-convex) with respect to 119877

+on the manifold 119876 = 120577 = (119911 119911) |

119911 isin 119862119899 sub 1198622119899 if for any 120577 = (119911 119911) 1205770= (1199110 1199110) isin 119876 one has

Re [119891 (119911 119911) minus 119891 (1199110 1199110)]

ge (gt) 119865 (119911 1199110 120572 (119911 119911

0)

times (nabla119911119891 (1199110 1199110) + nabla119911119891 (1199110 1199110))) + 120588120579

2

(119911 1199110)

(4)

Definition 5 The real part Re[119891] of analytic function 119891

119876 sub 1198622119899 rarr 119862 is said to be (119865 120572 120588 120579)-quasiconvex (strict(119865 120572 120588 120579)-quasiconvex) with respect to 119877

+on the manifold

119876 = 120577 = (119911 119911) | 119911 isin 119862119899 sub 1198622119899 if for any 120577 = (119911 119911)1205770= (1199110 1199110) isin 119876 one has

Re [119891 (119911 119911) minus 119891 (1199110 1199110)]

le (lt) 0 997904rArr 119865(119911 1199110 120572 (119911 119911

0)

times (nabla119911119891 (1199110 1199110) + nabla119911119891 (1199110 1199110)))

le minus1205881205792

(119911 1199110)

(5)

Definition 6 The real part Re[119891] of analytic function 119891

119876 sub 1198622119899 rarr 119862 is said to be (119865 120572 120588 120579)-pseudoconvex (strict(119865 120572 120588 120579)-pseudoconvex) with respect to119877

+on themanifold

119876 = 120577 = (119911 119911) | 119911 isin 119862119899 sub 1198622119899 if for any 120577 = (119911 119911)1205770= (1199110 1199110) isin 119876 one has

119865 (119911 1199110 120572 (119911 119911

0) (nabla119911119891 (1199110 1199110) + nabla119911119891 (1199110 1199110)))

ge minus1205881205792

(119911 1199110) 997904rArr Re [119891 (119911 119911) minus 119891 (119911

0 1199110)]

ge (gt) 0

(6)

Definition 7 The mapping ℎ 1198622119899 rarr 119862119901 is said to be(119865 120572 120588 120579)-convex (strict (119865 120572 120588 120579)-convex) with respect tothe polyhedral cone 119878 sub 119862

119901 on the manifold 119876 if for any120583 isin 119878 and 120577 = (119911 119911) 120577

0= (1199110 1199110) isin 119876 one has

Re ⟨120583 ℎ (120577) minus ℎ (1205770)⟩

ge (gt) 119865 (119911 1199110 120572 (119911 119911

0)

times (120583119879

nabla119911ℎ (1205770) + 120583119867

nabla119911ℎ (1205770))) + 120588120579

2

(119911 1199110)

(7)

Definition 8 The mapping ℎ 1198622119899 rarr 119862119901 is said to be(119865 120572 120588 120579)-quasiconvex (strict (119865 120572 120588 120579)-quasiconvex) with

respect to the polyhedral cone 119878 sub 119862119901 on the manifold 119876 if

for any 120583 isin 119878 and 120577 = (119911 119911) 1205770= (1199110 1199110) isin 119876 one has

Re ⟨120583 ℎ (120577) minus ℎ (1205770)⟩

le (lt) 0 997904rArr 119865(119911 1199110 120572 (119911 119911

0)

times (120583119879

nabla119911ℎ (1205770) + 120583119867

nabla119911ℎ (1205770)))

le minus1205881205792

(119911 1199110)

(8)

Definition 9 The mapping ℎ 1198622119899 rarr 119862119901 is said to be(119865 120572 120588 120579)-pseudoconvex (strict (119865 120572 120588 120579)-pseudoconvex)with respect to the polyhedral cone 119878 sub 119862

119901 on the manifold119876 if for any 120583 isin 119878 and 120577 = (119911 119911) 120577

0= (1199110 1199110) isin 119876 one has

119865(119911 1199110 120572 (119911 119911

0) (120583119879

nabla119911ℎ (1205770) + 120583119867

nabla119911ℎ (1205770)))

ge 1205881205792

(119911 1199110) 997904rArr Re ⟨120583 ℎ (120577) minus ℎ (120577

0)⟩

ge (gt) 0

(9)

Remark 10 In the proofs of theorems sometimes it may bemore convenient to use certain alternative but equivalentforms of the above definitions Consider the following exam-ple

The real part Re[119891] of analytic function119891 119876 sub 1198622119899 rarr 119862

is said to be (119865 120572 120588 120579)-pseudoconvex with respect to 119877+on

the manifold 119876 = 120577 = (119911 119911) | 119911 isin 119862119899 sub 1198622119899 if for any1205770= (1199110 1199110) isin 119876 one has

Re [119891 (119911 119911) minus 119891 (1199110 1199110)]

lt 0 997904rArr 119865(119911 1199110 120572 (119911 119911

0)

times (nabla119911119891 (1199110 1199110) + nabla119911119891 (1199110 1199110)))

lt minus1205881205792

(119911 1199110)

(10)

Remark 11 If we take 120572(119911 1199110) = 1 then the above definitions

reduce to that given by Lai et al [9] In addition if we take120588 = 0 then we obtain the definitions given by Mishra andRueda [17]

Let 119860 isin 119862119899times119899 and 119911 119906 isin 119862119899 then Schwarz inequality can

be written as

Re (119911119867119860119906) le (119911119867

119860119911)12

(119906119867

119860119906)12

(11)

The equality holds if 119860119911 = 120582119860119906 or 119911 = 120582119906 for 120582 ge 0

Definition 12 (see [12]) The problem (P) is said to satisfy theconstraint qualification at a point 120577

0= (1199110 1199110) if for any

nonzero 120583 isin 119878lowast sub 119862119901

Re ⟨ℎ1015840120577(1205770) (120577 minus 120577

0) 120583⟩ = 0 for 120577 = 120577

0 (12)

In the next section we recall some notations and discussnecessary and sufficient optimality conditions for problem(P) on the basis of Lai and Liu [18] and Lai and Huang [12]

4 Journal of Optimization

3 Necessary and Sufficient Conditions

Let 119891(120577 sdot) 120577 = (119911 119911) isin 1198622119899 be a continuous function definedon119884 where119884 sub 1198622119898 is a specified compact subset in problem(P) Then the supremum sup]isin119884 Re119891(120577 ]) will be attained toits maximum in 119884 and the set

119884 (120577) = 120578 isin 119884 | Re119891 (120577 120578) = sup]isin119884

Re119891 (120577 ]) (13)

is then also a compact set in 1198622119898 In particular if 120577 = 1205770

=

(1199110 1199110) is an optimal solution of problem (P) there exist a

positive integer 119896 and finite points 120578119894isin 119884(120577

0) 120582119894gt 0 119894 =

1 2 119896 withsum119896

119894=1120582119894= 1 such that the Lagrangian function

120601 (120577) =

119896

sum119894=1

120582119894119891 (120577 120578

119894) + ⟨ℎ (120577) 120583⟩ (120583 = 0 in 119878

lowast

) (14)

satisfies the Kuhn-Tucker type condition at 1205770 That is

(

119896

sum119894=1

1205821198941198911015840

120577(1205770 120578119894) + ⟨ℎ

1015840

120577(1205770) 120583⟩) (120577 minus 120577

0) = 0 (15)

Re ⟨ℎ (1205770) 120583⟩ = 0 (16)

Equivalent form of expression (15) at 120577 = 1205770isin 119876 is

119896

sum119894=1

120582119894[nabla119911119891 (1205770 120578119894) + nabla119911119891 (1205770 120578119894)]

+ (120583119879

nabla119911ℎ (1205770) + 120583119867

nabla119911ℎ (1205770)) = 0

(17)

For the integer 119896 corresponding a vector 120578 equiv (1205781 1205782 120578

119896) isin

119884(1205770)119896 and 120582

119894gt 0 119894 = 1 2 119896 withsum

119896

119894=1120582119894= 1 we define a

set as follows

119885120578(1205770) =

120577 isin 1198622119899

| minusℎ1015840

120577(1205770) 120577 isin 119878 (minusℎ (120577

0))

120577 = (119911 119911) isin 119876

Re[119896

sum119894=1

1205821198941198911015840

120577(1205770 120578119894) 120577 + ⟨119860119911 119911⟩

12

] lt 0

(18)

where the set 119878(1199040) is the intersection of closed half-spaces

having the point 1199040isin 119878 on their boundaries

Theorem 13 (necessary optimality conditions) Let 1205770

=

(1199110 1199110) isin 119876 be an optimal solution to (P) Suppose that the

constraint qualification is satisfied for (P) at 1205770and 119911119867

01198601199110

=

⟨1198601199110 1199110⟩ gt 0 Then there exist 0 = 120583 isin 119878lowast sub 119862119901 119906 isin 119862119899 and a

positive integer 119896 with the following properties

(i) 120578119894isin 119884(1205770) 119894 = 1 2 119896

(ii) 120582119894gt 0 119894 = 1 2 119896 sum119896

119894=1120582119894= 1

such that sum119896

119894=1120582119894119891(120577 120578

119894) + ⟨ℎ(120577) 120583⟩ + ⟨119860119911 119911⟩

12 satisfies thefollowing conditions

119896

sum119894=1

120582119894nabla119911119891 (1205770 120578119894) + nabla119911119891 (1205770 120578119894) + 119860119906

+ (120583119879

nabla119911ℎ (1205770) + 120583119867

nabla119911ℎ (1205770)) = 0

(19)

Re ⟨ℎ (1205770) 120583⟩ = 0 (20)

119906119867

119860119906 le 1 (21)

(119911119867

01198601199110)12

= Re (1199111198670119860119906) (22)

Theorem 14 (sufficient optimality conditions) Let 1205770

=

(1199110 1199110) isin 119876 be a feasible solution to (P) Suppose that there

exists a positive integer 119896 120582119894gt 0 120578119894isin 119884(1205770) 119894 = 1 2 119896with

sum119896

119894=1120582119894= 1 and 0 = 120583 isin 119878lowast sub 119862119901 satisfying conditions (19)ndash

(22) Further if Re[sum119896119894=1

120582119894[119891(120577 120578

119894) + 119911119867119860119906]] is (119865 120572

1 1205881 120579)-

convex with respect to 119877+on 119876 ℎ(120577) is (119865 120572

2 1205882 120579)-convex on

119876with respect to the polyhedral cone 119878 sub 119862119901 and12058811205721(119911 1199110)+

12058821205722(119911 1199110) ge 0 then 120577

0= (1199110 1199110) is an optimal solution to

(P)

Proof We prove this theorem by contradiction Suppose thatthere is a feasible solution 120577 isin 119876 such that

sup120578isin119884

Re [119891 (120577 120578) + (119911119867

119860119911)12

]

lt sup120578isin119884

Re [119891 (1205770 120578) + (119911

119867

01198601199110)12

]

(23)

Since 120578119894isin 119884(1205770) 119894 = 1 2 119896 we have

sup120578isin119884

Re [119891 (1205770 120578) + (119911

119867

01198601199110)12

]

= Re [119891 (1205770 120578119894) + (119911

119867

01198601199110)12

]

for 119894 = 1 2 119896

Re [119891 (120577 120578119894) + (119911

119867

119860119911)12

]

lt sup120578isin119884

Re [119891 (120577 120578) + (119911119867

119860119911)12

]

for 119894 = 1 2 119896

(24)

Thus from the above three inequalities we obtain

Re [119891 (120577 120578119894) + (119911

119867

119860119911)12

]

lt Re [119891 (1205770 120578119894) + (119911

119867

01198601199110)12

]

for 119894 = 1 2 119896

(25)

Journal of Optimization 5

Using (21) and generalized Schwarz inequality we get

Re (119911119867119860119906) le (119911119867

119860119911)12

(119906119867

119860119906)12

le (119911119867

119860119911)12

= Re [(119911119867119860119911)12

]

(26)

and inequality (22) yields

Re (1199111198670119860119906) = Re [(119911119867

01198601199110)12

] (27)

Using (26) and (27) in (25) we have

Re [119891 (120577 120578119894) + 119911119867

119860119906] lt Re [119891 (1205770 120578119894) + 119911119867

0119860119906]

for 119894 = 1 2 119896

(28)

Since 120582119894gt 0 and sum

119896

119894=1120582119894= 1 we have

119903119897Re[119896

sum119894=1

120582119894[119891 (120577 120578

119894) + 119911119867

119860119906] minus

119896

sum119894=1

120582119894[119891 (1205770 120578119894) + 119911119867

0119860119906]]

lt 0

(29)

Since Re[sum119896119894=1

120582119894[119891(120577 120578

119894) + 119911119867119860119906]] is (119865 120572

1 1205881 120579)-convex

with respect to 119877+on 119876 we have

Re[119896

sum119894=1

120582119894[119891 (120577 120578

119894) + 119911119867

119860119906]

minus

119896

sum119894=1

120582119894[119891 (1205770 120578119894) + 119911119867

0119860119906]]

ge 119865[119911 1199110 1205721(119911 1199110)

times(

119896

sum119894=1

120582119894nabla119911119891 (1205770 120578119894) + nabla119911119891 (1205770 120578119894) + 119860119906)]

+ 12058811205792

(119911 1199110)

(30)

From (29) and (30) we conclude that

119865[119911 1199110 1205721(119911 1199110)

times(

119896

sum119894=1

120582119894nabla119911119891(1205770 120578119894) + nabla119911119891 (1205770 120578119894) + 119860119906)]

lt minus12058811205792

(119911 1199110)

(31)

which due to sublinearity of 119865 can be written as

119865[119911 1199110

119896

sum119894=1

120582119894nabla119911119891 (1205770 120578119894) + nabla119911119891 (1205770 120578119894) + 119860119906]

lt minus1205881

1205721(119911 1199110)1205792

(119911 1199110)

(32)

On the other hand from the feasibility of 120577 to (P) we haveminusℎ(120577) isin 119878 or Re⟨ℎ(120577) 120583⟩ le 0 for 120583 isin 119878lowast which along with(20) yields

Re ⟨ℎ (120577) 120583⟩ le 0 = Re ⟨ℎ (1205770) 120583⟩ (33)

Since ℎ(120577) is (119865 1205722 1205882 120579)-convex on 119876 with respect to the

polyhedral cone 119878 sub 119862119901 we have

Re ⟨ℎ (120577) 120583⟩ minus Re ⟨ℎ (1205770) 120583⟩

ge 119865 [119911 1199110 1205722(119911 1199110) (120583119879

nabla119911ℎ (1205770) + 120583119867

nabla119911ℎ (1205770))]

+ 12058821205792

(119911 1199110)

(34)

From (33) and (34) it follows that

119865 [119911 1199110 1205722(119911 1199110) (120583119879

nabla119911ℎ (1205770) + 120583119867

nabla119911ℎ (1205770))]

le minus12058821205792

(119911 1199110)

(35)

which due to sublinearity of 119865 can be written as

119865 [119911 1199110 120583119879

nabla119911ℎ (1205770) + 120583119867

nabla119911ℎ (1205770)] le minus

1205882

1205722(119911 1199110)1205792

(119911 1199110)

(36)

On adding (32) and (36) and using sublinearity of 119865 we get

119865[119911 1199110

119896

sum119894=1

120582119894nabla119911119891 (1205770 120578119894) + nabla119911119891 (1205770 120578119894) + 119860119906

+120583119879

nabla119911ℎ (1205770) + 120583119867

nabla119911ℎ (1205770) ]

lt minus(1205881

1205721(119911 1199110)+

1205882

1205722(119911 1199110)) 1205792

(119911 1199110)

(37)

The above inequality together with the assumption12058811205721(119911 1199110) + 12058821205722(119911 1199110) ge 0 gives

119865[119911 1199110

119896

sum119894=1

120582119894nabla119911119891 (1205770 120578119894) + nabla119911119891 (1205770 120578119894) + 119860119906

+ 120583119879

nabla119911ℎ (1205770) + 120583119867

nabla119911ℎ (1205770) ] lt 0

(38)

which contradicts (19) hence the theorem

6 Journal of Optimization

Theorem 15 (sufficient optimality conditions) Let 1205770

=

(1199110 1199110) isin 119876 be a feasible solution to (P) Suppose that there

exists a positive integer 119896 120582119894gt 0 120578119894isin 119884(1205770) 119894 = 1 2 119896with

sum119896

119894=1120582119894= 1 and 0 = 120583 isin 119878lowast sub 119862119901 satisfying conditions (19)ndash

(22) Further if Re[sum119896119894=1

120582119894[119891(120577 120578

119894) + 119911119867119860119906]] is (119865 120572

1 1205881 120579)-

pseudoconvex with respect to 119877+on 119876 ℎ(120577) is (119865 120572

2 1205882 120579)-

quasiconvex on 119876 with respect to the polyhedral cone 119878 sub 119862119901and 120588

11205721(119911 1199110) + 12058821205722(119911 1199110) ge 0 then 120577

0= (1199110 1199110) is an

optimal solution to (P)

Proof Proceeding as in Theorem 14 we have

Re[119896

sum119894=1

120582119894[119891 (120577 120578

119894) + 119911119867

119860119906] minus

119896

sum119894=1

120582119894[119891 (1205770 120578119894) + 119911119867

0119860119906]]

lt 0

(39)

which by (119865 1205721 1205881 120579)-pseudoconvexity of

Re[sum119896119894=1

120582119894[119891(120577 120578

119894) + 119911119867119860119906]] with respect to 119877

+on 119876

yields

119865[119911 1199110 1205721(119911 1199110)

119896

sum119894=1

120582119894nabla119911119891 (1205770 120578119894) + nabla119911119891 (1205770 120578119894) + 119860119906]

lt minus12058811205792 (119911 119911

0)

(40)

Using the sublinearity of 119865 the above inequality can bewritten as

119865[119911 1199110

119896

sum119894=1

120582119894nabla119911119891 (1205770 120578119894) + nabla119911119891 (1205770 120578119894) + 119860119906]

lt minus1205881

1205721(119911 1199110)1205792 (119911 119911

0)

(41)

On the other hand from the feasibility of 120577 to (P) we haveminusℎ(120577) isin 119878 or Re⟨ℎ(120577) 120583⟩ le 0 for 120583 isin 119878lowast which along with(20) yields

Re ⟨ℎ (120577) 120583⟩ le 0 = Re ⟨ℎ (1205770) 120583⟩ (42)

Since ℎ(120577) is (119865 1205722 1205882 120579)-quasiconvex on 119876 with respect to

the polyhedral cone 119878 sub 119862119901 the above inequality yields

119865 [119911 1199110 1205722(119911 1199110) (120583119879

nabla119911ℎ (1205770) + 120583119867

nabla119911ℎ (1205770))]

le minus12058821205792 (119911 119911

0)

(43)

which due to sublinearity of 119865 can be written as

119865 [119911 1199110 120583119879

nabla119911ℎ (1205770) + 120583

119867

nabla119911ℎ (1205770)] le minus

1205882

1205722(119911 1199110)1205792

(119911 1199110)

(44)

On adding (41) and (44) and using sublinearity of 119865 we get

119865[119911 1199110

119896

sum119894=1

120582119894nabla119911119891 (1205770 120578119894) + nabla119911119891 (1205770 120578119894) + 119860119906

+ 120583119879

nabla119911ℎ (1205770) + 120583119867

nabla119911ℎ (1205770) ]

lt minus(1205881

1205721(119911 1199110)+

1205882

1205722(119911 1199110)) 1205792

(119911 1199110)

(45)

The above inequality together with the assumption12058811205721(119911 1199110) + 12058821205722(119911 1199110) ge 0 gives

119865[119911 1199110

119896

sum119894=1

120582119894nabla119911119891 (1205770 120578119894) + nabla119911119891 (1205770 120578119894) + 119860119906

+120583119879

nabla119911ℎ (1205770) + 120583119867

nabla119911ℎ (1205770) ] lt 0

(46)

which contradicts (19) hence the theorem

Theorem 16 (sufficient optimality conditions) Let 1205770

=

(1199110 1199110) isin 119876 be a feasible solution to (P) Suppose that there

exists a positive integer 119896 120582119894gt 0 120578119894isin 119884(1205770) 119894 = 1 2 119896with

sum119896

119894=1120582119894= 1 and 0 = 120583 isin 119878lowast sub 119862119901 satisfying conditions (19)ndash

(22) Further if Re[sum119896119894=1

120582119894[119891(120577 120578

119894) + 119911119867119860119906]] is (119865 120572

1 1205881 120579)-

quasiconvex with respect to 119877+on119876 ℎ(120577) is strict (119865 120572

2 1205882 120579)-

pseudoconvex on119876 with respect to the polyhedral cone 119878 sub 119862119901

and 12058811205721(119911 1199110) + 12058821205722(119911 1199110) ge 0 then 120577

0= (1199110 1199110) is an

optimal solution to (P)

Proof The proof follows on the similar lines of Theorem 15

4 Parametric Duality

We adopt the following notations in order to simplify theformulation of dual

119870 (120585) = (119896 120578) isin 119873 times 119877119896

+times 1198622119898119896

|

= (1205821 1205822 120582

119896) with

119896

sum119894=1

120582119894= 1

120578 = (1205781 1205782 120578

119896)

with 120578119894isin 119884 (120585) 119894 = 1 2 119896

(47)

for 120585 = (119906 119906) isin 119876 sub 1198622119899Now we formulate a parametric dual problem (D1) with

respect to the complex minimax programming problem (P)as follows

max(119896120582120578)isin119870(120585)

sup(120585120583119908119905)isin119883(119896120582120578)

119905(D1)

Journal of Optimization 7

where119883(119896 120578) denotes the set of all (120585 120583 119908 119905) isin 1198622119899 times119862119901 times

119862119899 times 119877 to satisfy the following conditions119896

sum119894=1

120582119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]

+ 120583119879 nabla119911ℎ (120585) + 120583119867nabla

119911ℎ (120585) = 0

(48)

119896

sum119894=1

120582119894Re119891 (120585 120578

119894) + (119906

119867

119860119906)12

minus 119905 ge 0 (49)

Re ⟨ℎ (120585) 120583⟩ ge 0 (50)

119908119867

119860119908 le 1 (51)

(119906119867

119860119906)12

= Re (119906119867119860119908) (52)

0 = 120583 isin 119878lowast

(53)

If for a triplet (119896 120578) isin 119870(120585) the set 119883(119896 120578) = 0then we define the supremum over 119883(119896 120578) to be minusinfin fornonexecption in the formulation of (D1)

Theorem 17 (weak duality) Let 120577 = (119911 119911) and(119896 120578 120585 120583 119908 119905) be feasible solutions to (P) and (D1)respectively Further if Re[sum119896

119894=1120582119894[119891(120577 120578

119894) + 119911119867119860119908]] is

(119865 1205721 1205881 120579)-pseudoconvex with respect to 119877

+on 119876 ℎ(120577) is

(119865 1205722 1205882 120579)-quasiconvex on 119876 with respect to the polyhedral

cone 119878 sub 119862119901 and 12058811205721(119911 119906) + 120588

21205722(119911 119906) ge 0 Then

sup120578isin119884

Re [119891 (120577 120578) + (119911119867

119860119911)12

] ge 119905 (54)

Proof Suppose on the contrary that

sup120578isin119884

Re [119891 (120577 120578) + (119911119867

119860119911)12

] lt 119905 (55)

By compactness of 119884(120585) sub 119884 in 119862119901 120585 isin (119906 119906) isin 119876 thereexist an integer 119896 gt 0 and finite points 120578

119894isin 119884(120585) 120582

119894gt 0

119894 = 1 2 119896 with sum119896

119894=1120582119894= 1 such that (49) holds From

(49) and (55) we have

Re[119896

sum119894=1

120582119894[119891 (120577 120578

119894) + (119911

119867

119860119911)12

]] lt

119896

sum119894=1

120582119894119905

le

119896

sum119894=1

120582119894Re [119891 (120585 120578

119894) + (119906

119867

119860119906)12

]

(56)

From (51) and the generalized Schwarz inequality we have

Re (119911119867119860119908) le (119911119867

119860119911)12

(119908119867

119860119908)12

le (119911119867

119860119911)12

(57)

Using (52) and (57) in (56) we get

Re[119896

sum119894=1

120582119894[119891 (120577 120578

119894) + 119911119867

119860119908]]

lt Re[119896

sum119894=1

120582119894[119891 (120585 120578

119894) + 119906119867

119860119908]]

(58)

Since Re[sum119896119894=1

120582119894[119891(120577 120578

119894) + 119911119867119860119908]] is (119865 120572

1 1205881 120579)-

pseudoconvex with respect to 119877+on 119876 the above inequality

implies that

119865[119911 119906 1205721(119911 119906)

119896

sum119894=1

120582119894nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]

lt minus12058811205792

(119911 119906)

(59)

which due to sublinearity of 119865 can be written as

119865[119911 119906

119896

sum119894=1

120582119894nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]

lt minus1205881

1205721(119911 119906)

1205792

(119911 119906)

(60)

By the feasibility of 120577 = (119911 119911) to (P) we have minusℎ(120577) isin 119878 orRe⟨ℎ(120577) 120583⟩ le 0 for 120583 isin 119878lowast which along with (50) yields

Re ⟨ℎ (120577) 120583⟩ le 0 le Re ⟨ℎ (120585) 120583⟩ (61)

The above inequality together with the (119865 1205722 1205882 120579)-

quasiconvexity of ℎ(120577) on 119876 with respect to the polyhedralcone 119878 sub 119862

119901 implies

119865 [119911 119906 1205722(119911 119906) (120583

119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

le minus12058821205792

(119911 119906)

(62)

which due to sublinearity of 119865 can be written as

119865 [119911 119906 120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585)]

le minus1205882

1205722(119911 119906)

1205792

(119911 119906) (63)

On adding (60) and (63) and using sublinearity of 119865 we get

119865[119911 119906

119896

sum119894=1

120582119894nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908

+ 120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585) ]

lt minus(1205881

1205721(119911 119906)

+1205882

1205722(119911 119906)

) 1205792

(119911 119906)

(64)

From the assumption 12058811205721(119911 119906) +120588

21205722(119911 119906) ge 0 the above

inequality yields

119865[119911 119906

119896

sum119894=1

120582119894nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908

+ 120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585) ] lt 0

(65)

which contradicts (48) hence the theorem

8 Journal of Optimization

Theorem 18 (weak duality) Let 120577 = (119911 119911) and(119896 120578 120585 120583 119908 119905) be feasible solutions to (P) and (D1)respectively Further if Re[sum119896

119894=1120582119894[119891(120577 120578

119894) + 119911

119867

119860119908]] is(119865 1205721 1205881 120579)-quasiconvex with respect to 119877

+on 119876 ℎ(120577) is

(119865 1205722 1205882 120579)-pseudoconvex on 119876 with respect to the polyhedral

cone 119878 sub 119862119901 and 12058811205721(119911 119906) + 120588

21205722(119911 119906) ge 0 then

sup120578isin119884

Re [119891 (120577 120578) + (119911119867

119860119911)12

] ge 119905 (66)

Proof Theproof follows the same lines as inTheorem 17

Theorem 19 (strong duality) Let 1205770= (1199110 1199110) be an optimal

solution to the problem (P) at which a constraint qualificationis satisfied Then there exist (119896 120578) isin 119870(120577

0) and (120577

0 120583 119908 119905) isin

119883(119896 120578) such that (119896 120578 1205770 120583 119908 119905) is a feasible solution to

the dual problem (D1) If the hypotheses of Theorem 17 or 18are satisfied then (119896 120578 120577

0 120583 119908 119905) is optimal to (D1) and

the two problems (P) and (D1) have the same optimal values

Proof Theproof follows along the lines ofTheorem6 (Lai andLiu [13])

Theorem 20 (strict converse duality) Let 120577 and(

120582 120578 120585 120583 119908 ) be optimal solutions to (P) and (D1)

respectively and assume that the assumptions of Theorem 19are satisfied Further assume that the following conditions aresatisfied

(i) Resum119896

119894=1119894[119891(120577 120578

119894) + 119867119860119908] is strict (119865 120572

1 1205881 120579)-

pseudoconvex with respect to 119877+on 119876 and ℎ(120577) is

(119865 1205722 1205882 120579)-quasiconvex on 119876 with respect to the

polyhedral cone 119878 sub 119862119901(ii) 12058811205721( ) + 120588

21205722( ) ge 0

Then 120577 = 120585 that is 120577 is optimal solution to (D1)

Proof On the contrary suppose that ( ) = 120577 = 120585 = ( )On applyingTheorem 19 we know that

= supisin119884

Re [119891 (120577 120578) + (119867

119860)12

] (67)

From the feasibility of 120577 isin 119876 to (P) 120583 isin 119878lowast and (50) we have

Re ⟨ℎ (120577) 120583⟩ le 0 le Re ⟨ℎ (120585) 120583⟩ (68)

Since ℎ(120577) is (119865 1205722 1205882 120579)-quasiconvex on 119876 with respect to

the polyhedral cone Sin 119862119901 the above inequality yields

119865 [ 1205722( ) (120583

119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

le minus12058821205792

( )

(69)

which by sublinearity of 119865 implies

119865 [ (120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

le minus1205882

1205722( )

1205792

( )

(70)

By (48) and the sublinearity of 119865 we have

119865[

[

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]]

]

+ 119865 [ (120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

ge 119865[

[

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]

+ 120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585)]

]

= 0

(71)

The above inequality together with (70) and 12058811205721( ) +

12058821205722( ) ge 0 gives

119865[

[

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]]

]

ge minus119865 [ (120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

ge1205882

1205722( )

1205792

( )

ge minus1205881

1205721( )

1205792

( )

(72)

That is

119865[

[

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]]

]

ge minus1205881

1205721( )

1205792

( )

(73)

which by sublinearity of 119865 implies

119865[

[

1205721( )

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]]

]

ge minus12058811205792

( )

(74)

Since Resum119896

119894=1119894[119891(120577 120578

119894) + 119867119860119908] is strict (119865 120572

1 1205881 120579)-

pseudoconvex with respect to 119877+on 119876 the above inequality

implies that

Re[

[

119896

sum119894=1

119894[119891 (120577 120578

119894) + 119867

119860119908]]

]

gt Re[

[

119896

sum119894=1

119894[119891 (120585 120578

119894) + 119867

119860119908]]

]

(75)

Journal of Optimization 9

From (51) (52) and the generalized Schwarz inequality wehave

Re (119867119860119908) le (119867

119860)12

Re (119867119860119908) = (119867

119860)12

(76)

which on substituting in (75) and by using (49) we obtain

Re[

[

119896

sum119894=1

119894[119891 (120577 120578

119894) + (

119867

119860)12

]]

]

gt Re[

[

119896

sum119894=1

119894[119891 (120585 120578

119894) + (

119867

119860)12

]]

]

ge

119896

sum119894=1

119894

(77)

Consequently there exist certain 1198940which satisfy

Re [119891 (120577 1205781198940) + (

119867

119860)12

] gt (78)

Hence

supisin119884

Re [119891 (120577 120578) + (119867

119860)12

]

ge Re [119891 (120577 1205781198940) + (

119867

119860)12

] gt

(79)

which contradicts (67) hence the theorem

5 Parameter Free Duality

Making use of the optimality conditions we show thatthe following formation is a dual (D2) to the complexprogramming problem (P)

max(119896120582120578)isin119870(120585)

sup(120585120583119908)isin119883(119896120582120578)

Re [119891 (120585 120578) + (119906119867

119860119906)12

] (D2)

where119883(119896 120578) denotes the set of all (120585 120583 119908) isin 1198622119899times119862119901times119862119899

to satisfy the following conditions

119896

sum119894=1

120582119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]

+ 120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585) = 0

(80)

Re ⟨ℎ (120585) 120583⟩ ge 0 (81)

119908119867

119860119908 le 1 (82)

(119906119867

119860119906)12

= Re (119906119867119860119908) (83)

If for a triplet (119896 120578) isin 119870(120585) the set 119883(119896 120578) = 0then we define the supremum over 119883(119896 120578) to be minusinfin fornonexception in the formulation of (D2)

Nowwe establish appropriate duality theorems and provethat optimal values of (P) and (D2) are equal under theassumption of generalized convexity in order to show that theproblems (P) and (D2) have no duality gap

Theorem21 (weak duality) Let 120577 = (119911 119911) and (119896 120578 120585 120583 119908)

be feasible solutions to (P) and (D2) respectively Furtherif Re[sum119896

119894=1120582119894[119891(120577 120578

119894) + 119911119867119860119908]] is (119865 120572

1 1205881 120579)-pseudoconvex

with respect to 119877+on 119876 ℎ(120577) is (119865 120572

2 1205882 120579)-quasiconvex on 119876

with respect to the polyhedral cone 119878 sub 119862119901 and 12058811205721(z 119906) +

12058821205722(119911 119906) ge 0 then

sup120578isin119884

Re [119891 (120577 120578) + (119911119867

119860119911)12

]

ge sup120578isin119884

Re [119891 (120585 120578) + (119906119867

119860119906)12

]

(84)

Proof On the contrary we suppose that

sup120578isin119884

Re [119891 (120577 120578) + (119911119867

119860119911)12

]

lt sup120578isin119884

Re [119891 (120585 120578) + (119906119867

119860119906)12

]

(85)

Since 120578119894isin 119884(120585) sub 119884 119894 = 1 2 119896 we have

sup120578isin119884

Re [119891 (120585 120578) + (119906119867

119860119906)12

]

= Re [119891 (120585 120578119894) + (119906

119867

119860119906)12

] 119894 = 1 2 119896

(86)

Re [119891 (120577 120578119894) + (119911

119867

119860119911)12

]

le sup120578isin119884

Re [119891 (120577 120578) + (119911119867

119860119911)12

] 119894 = 1 2 119896

(87)

Then the above three inequalities give

Re [119891 (120577 120578119894) + (119911

119867

119860119911)12

]

lt Re [119891 (120585 120578119894) + (119906

119867

119860119906)12

] 119894 = 1 2 119896

(88)

From (82) (83) (88) and the generalized Schwarz inequalitywe have

Re [119891 (120577 120578119894) + (119911

119867

119860119908)]

lt Re [119891 (120585 120578119894) + (119906

119867

119860119908)] 119894 = 1 2 119896

(89)

As 120582119894gt 0 119894 = 1 2 119896 and sum

119896

119894=1120582119894= 1 we have

Re[119896

sum119894=1

120582119894[119891 (120577 120578

119894) + (119911

119867

119860119908)]]

minus Re[119896

sum119894=1

120582119894[119891 (120585 120578

119894) + (119906

119867

119860119908)]] lt 0

(90)

10 Journal of Optimization

Since Re[sum119896119894=1

120582119894[119891(120577 120578

119894) + 119911119867119860119908]] is (119865 120572

1 1205881 120579)-

pseudoconvex with respect to 119877+on 119876 the above inequality

implies that

119865[119911 119906 1205721(119911 119906)

times

119896

sum119894=1

120582119894nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]

lt minus12058811205792

(119911 119906)

(91)

which by sublinearity of 119865 becomes

119865[119911 119906

119896

sum119894=1

120582119894nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]

lt minus1205881

1205721(119911 119906)

1205792

(119911 119906)

(92)

By the feasibility of 120577 = (119911 119911) to (P) 0 = 120583 isin 119878lowast and theinequality (81) we obtain

Re ⟨ℎ (120577) 120583⟩ le 0 le Re ⟨ℎ (120585) 120583⟩ (93)

The above inequality together with the (119865 1205722 1205882 120579)-

quasiconvexity of ℎ(120577) on 119876 with respect to the polyhedralcone 119878 sub 119862

119901 implies

119865 [119911 119906 1205722(119911 119906) (120583

119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

le minus12058821205792

(119911 119906)

(94)

which by sublinearity of 119865 becomes

119865 [119911 119906 120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585)]

le minus1205882

1205722(119911 119906)

1205792

(119911 119906) (95)

On adding (92) and (95) and using the sublinearity of 119865 weget

119865[119911 119906

119896

sum119894=1

120582119894nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908

+120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585) ]

lt minus(1205881

1205721(119911 119906)

+1205882

1205722(119911 119906)

) 1205792

(119911 119906)

(96)

From the assumption 12058811205721(119911 119906) +120588

21205722(119911 119906) ge 0 the above

inequality yields

119865[119911 119906

119896

sum119894=1

120582119894nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908

+ 120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585) ] lt 0

(97)

which contradicts (80) hence the theorem

Theorem 22 (weak duality) Let 120577 = (119911 119911) and(119896 120578 120585 120583 119908) be feasible solutions to (P) and (D2)respectively Further if Re[sum119896

119894=1120582119894[119891(120577 120578

119894) + 119911

119867

119860119908]] is(119865 1205721 1205881 120579)-quasiconvex with respect to 119877

+on 119876 ℎ(120577) is

(119865 1205722 1205882 120579)-pseudoconvex on 119876 with respect to the polyhedral

cone 119878 sub 119862119901 and 12058811205721(119911 119906) + 120588

21205722(119911 119906) ge 0 then

sup120578isin119884

Re [119891 (120577 120578) + (119911119867

119860119911)12

]

ge sup120578isin119884

Re [119891 (120585 120578) + (119906119867

119860119906)12

]

(98)

Proof Theproof follows the same lines as inTheorem 21

Theorem 23 (strong duality) Let 1205770= (1199110 1199110) be an optimal

solution to the problem (P) at which a constraint qualificationis a satisfied Then there exist (119896 120578) isin 119870(120577

0) and (120577

0 120583 119908) isin

119883(119896 120578) such that (119896 120578 1205770 120583 119908) is a feasible solution to the

dual problem (D2) Further if the hypotheses of Theorem 21or Theorem 22 are satisfied then (119896 120578 120577

0 120583 119908) is optimal

to (D2) and the two problems (P) and (D2) have the sameoptimal values

Proof Theproof follows along the lines ofTheorem8 (Lai andLiu [13])

Theorem 24 (strict converse duality) Let 120577 and(

120582 120578 120585 120583 119908) be optimal solutions to (P) and (D2)

respectively and the conditions of Theorem 23 are satisfiedFurther assume that the following conditions are satisfied

(i) Resum119896

119894=1119894[119891(120577 120578

119894) + 119911119867119860119908] is strict (119865 120572

1 1205881 120579)-

pseudoconvex with respect to 119877+on 119876 and ℎ(120577) is

(119865 1205722 1205882 120579)-quasiconvex on 119876 with respect to the

polyhedral cone 119878 sub 119862119901(ii) 12058811205721( ) + 120588

21205722( ) ge 0

Then 120577 = 120585 that is 120585 is an optimal solution to (D2)

Proof On the contrary we assume that ( ) = 120577 = 120585 = ( )On applyingTheorem 23 we know that

supisin119884

Re [119891 (120577 120578) + (119867

119860)12

]

= supisin119884

Re [119891 (120585 120578) + (119867

119860)12

]

(99)

From the feasibility of 120577 isin 119876 to (P) 120583 isin 119878lowast inequality (81)yields

Re ⟨ℎ (120577) 120583⟩ le 0 le Re ⟨ℎ (120585) 120583⟩ (100)

Since ℎ(120577) is (119865 1205722 1205882 120579)-quasiconvex on 119876 with respect to

the polyhedral cone 119878 in 119862119901 the above inequality yields

119865 [ 1205722( ) (120583

119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

le minus12058821205792

( )

(101)

Journal of Optimization 11

which by sublinearity of 119865 implies

119865 [ (120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

le minus1205882

1205722( )

1205792

( )

(102)

By (80) and the sublinearity of 119865 we have

119865[

[

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]]

]

+ 119865 [ (120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

ge 119865[

[

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]

+ 120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585)]

]

= 0

(103)

The above inequality together with (102) and 12058811205721( ) +

12058821205722( ) ge 0 gives

119865[

[

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]]

]

ge minus119865 [ (120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

ge1205882

1205722( )

1205792

( )

ge minus1205881

1205721( )

1205792

( )

(104)

That is

119865[

[

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]]

]

ge minus1205881

1205721( )

1205792 ( )

(105)

which by sublinearity of 119865 implies

119865[

[

1205721( )

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]]

]

ge minus12058811205792 ( )

(106)

Since Resum119896

119894=1119894[119891(120577 120578

119894) + 119911119867119860119908] is strict (119865 120572

1 1205881 120579)-

pseudoconvex with respect to 119877+on 119876 the above inequality

implies that

Re[

[

119896

sum119894=1

119894[119891 (120577 120578

119894) + 119867

119860119908]]

]

gt Re[

[

119896

sum119894=1

119894[119891 (120585 120578

119894) + 119867

119860119908]]

]

(107)

From (82) (83) and the generalized Schwarz inequality weget

Re (119867119860119908) le (119867

119860)12

Re (119867119860119908)12

= (119867

119860)12

(108)

which on substituting in (107) we obtain

Re[

[

119896

sum119894=1

119894[119891 (120577 120578

119894) + (

119867

119860)12

] ]

]

gt Re[

[

119896

sum119894=1

119894[119891 (120585 120578

119894) + (

119867

119860)12

]]

]

(109)

Consequently there exist certain 1198940which satisfy

Re [119891 (120577 1205781198940) + (119867119860)

12

]

gt Re [119891 (120585 1205781198940) + (119867119860)

12

]

(110)

Hence

supisin119884

Re [119891 (120577 120578) + (119867

119860)12

]

ge Re [119891 (120577 1205781198940) + (

119867

119860)12

]

gt Re [119891 (120585 1205781198940) + (

119867

119860)12

]

= supisin119884

Re [119891 (120585 120578) + (119867

119860)12

]

(111)

which contradicts (99) hence the theorem

6 Conclusion

In this paper we introduced generalized (119865 120572 120588 120579)-convexfunctions and established sufficient optimality conditions fora class of nondifferentiable minimax programming problemsin complex space These optimality conditions are thenused to construct two types of dual model and finally wederived weak strong and strict converse duality theorems

12 Journal of Optimization

to show that there is no duality gap between the two dualproblems with respect to the primal problem under somegeneralized convexities of complex functions in the complexprogramming problem As a future task the authors wouldlike to extend these results to second and higher order casesand establish the relations between primal and its second andhigher order dual problems

Acknowledgments

The authors are thankful to the anonymous referees for theirvaluable comments which have improved the presentationof the paper The research of the first author is financiallysupported by the University Grant Commission New DelhiIndia through Grant No F no 41-8012012(SR)

References

[1] N Levinson ldquoLinear programming in complex spacerdquo Journalof Mathematical Analysis and Applications vol 14 no 1 pp 44ndash62 1966

[2] J C Chen and H C Lai ldquoOptimality conditions for minimaxprogramming of analytic functionsrdquoTaiwanese Journal ofMath-ematics vol 8 no 4 pp 673ndash686 2004

[3] N Datta and D Bhatia ldquoDuality for a class of nondifferentiablemathematical programming problems in complex spacerdquo Jour-nal of Mathematical Analysis and Applications vol 101 no 1 pp1ndash11 1984

[4] O P Jain and P C Saxena ldquoA duality theorem for a specialclass of programming problems in complex spacerdquo Journal ofOptimization Theory and Applications vol 16 no 3-4 pp 207ndash220 1975

[5] I M Stancu-Minasian Fractional Programming Theory Meth-ods and Applications vol 409 Kluwer Academic DordrechtGermany 1997

[6] I Ahmad and Z Husain ldquoOptimality conditions and dualityin nondifferentiable minimax fractional programming withgeneralized convexityrdquo Journal of Optimization Theory andApplications vol 129 no 2 pp 255ndash275 2006

[7] A Jayswal I Ahmad and D Kumar ldquoNondifferentiable mini-max fractional programming problem with nonsmooth gener-alized convex functionsrdquoCommunications onAppliedNonlinearAnalysis vol 18 no 4 pp 57ndash75 2011

[8] A Jayswal and D Kumar ldquoOn nondifferentiable minimaxfractional programming involving generalized (C 120572 120588 d)-convexityrdquo Communications on Applied Nonlinear Analysis vol18 no 1 pp 61ndash77 2011

[9] H-C Lai J-C Lee and S-C Ho ldquoParametric duality on mini-max programming involving generalized convexity in complexspacerdquo Journal of Mathematical Analysis and Applications vol323 no 2 pp 1104ndash1115 2006

[10] K Swarup and I C Sharma ldquoProgramming with linear frac-tional functionals in complex spacerdquo Cahiers Center drsquoEtudesRecherche Operationelle vol 12 pp 103ndash109 1970

[11] A Charnes and W W Cooper ldquoProgramming with linearfractional functionalsrdquo Naval Research Logistics Quarterly vol9 no 3-4 pp 181ndash186 1962

[12] H-C Lai and T-Y Huang ldquoOptimality conditions for non-differentiable minimax fractional programming with complexvariablesrdquo Journal of Mathematical Analysis and Applicationsvol 359 no 1 pp 229ndash239 2009

[13] H-C Lai and J-C Liu ldquoDuality for nondifferentiable minimaxprogramming in complex spacesrdquo Nonlinear Analysis TheoryMethods and Applications vol 71 no 12 pp e224ndashe233 2009

[14] I Ahmad Z Husain and S Sharma ldquoSecond-order dualityin nondifferentiable minmax programming involving type-Ifunctionsrdquo Journal of Computational and Applied Mathematicsvol 215 no 1 pp 91ndash102 2008

[15] I Ahmad Z Husain and S Sharma ldquoHigher-order duality innondifferentiableminimax programmingwith generalized typeI functionsrdquo Journal of Optimization Theory and Applicationsvol 141 no 1 pp 1ndash12 2009

[16] BMond andBD Craven ldquoA class of nondifferentiable complexprogramming problemsrdquo Journal of Mathematics and Statisticsvol 6 pp 581ndash591 1975

[17] S K Mishra and N G Rueda ldquoSymmetric duality for math-ematical programming in complex spaces with F-convexityrdquoJournal of Mathematical Analysis and Applications vol 284 no1 pp 250ndash265 2003

[18] H C Lai and J C Liu ldquoMinimax programming in complexspaces-necessary and sufficient optimality conditionsrdquo in Pro-ceedings of the International Symposium on Nonlinear Analysisand Convex Analysis 2007 Research Institute forMathematicalSciences Kyoto University Japan Rims Kokuroku 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Journal of Optimization

3 Necessary and Sufficient Conditions

Let 119891(120577 sdot) 120577 = (119911 119911) isin 1198622119899 be a continuous function definedon119884 where119884 sub 1198622119898 is a specified compact subset in problem(P) Then the supremum sup]isin119884 Re119891(120577 ]) will be attained toits maximum in 119884 and the set

119884 (120577) = 120578 isin 119884 | Re119891 (120577 120578) = sup]isin119884

Re119891 (120577 ]) (13)

is then also a compact set in 1198622119898 In particular if 120577 = 1205770

=

(1199110 1199110) is an optimal solution of problem (P) there exist a

positive integer 119896 and finite points 120578119894isin 119884(120577

0) 120582119894gt 0 119894 =

1 2 119896 withsum119896

119894=1120582119894= 1 such that the Lagrangian function

120601 (120577) =

119896

sum119894=1

120582119894119891 (120577 120578

119894) + ⟨ℎ (120577) 120583⟩ (120583 = 0 in 119878

lowast

) (14)

satisfies the Kuhn-Tucker type condition at 1205770 That is

(

119896

sum119894=1

1205821198941198911015840

120577(1205770 120578119894) + ⟨ℎ

1015840

120577(1205770) 120583⟩) (120577 minus 120577

0) = 0 (15)

Re ⟨ℎ (1205770) 120583⟩ = 0 (16)

Equivalent form of expression (15) at 120577 = 1205770isin 119876 is

119896

sum119894=1

120582119894[nabla119911119891 (1205770 120578119894) + nabla119911119891 (1205770 120578119894)]

+ (120583119879

nabla119911ℎ (1205770) + 120583119867

nabla119911ℎ (1205770)) = 0

(17)

For the integer 119896 corresponding a vector 120578 equiv (1205781 1205782 120578

119896) isin

119884(1205770)119896 and 120582

119894gt 0 119894 = 1 2 119896 withsum

119896

119894=1120582119894= 1 we define a

set as follows

119885120578(1205770) =

120577 isin 1198622119899

| minusℎ1015840

120577(1205770) 120577 isin 119878 (minusℎ (120577

0))

120577 = (119911 119911) isin 119876

Re[119896

sum119894=1

1205821198941198911015840

120577(1205770 120578119894) 120577 + ⟨119860119911 119911⟩

12

] lt 0

(18)

where the set 119878(1199040) is the intersection of closed half-spaces

having the point 1199040isin 119878 on their boundaries

Theorem 13 (necessary optimality conditions) Let 1205770

=

(1199110 1199110) isin 119876 be an optimal solution to (P) Suppose that the

constraint qualification is satisfied for (P) at 1205770and 119911119867

01198601199110

=

⟨1198601199110 1199110⟩ gt 0 Then there exist 0 = 120583 isin 119878lowast sub 119862119901 119906 isin 119862119899 and a

positive integer 119896 with the following properties

(i) 120578119894isin 119884(1205770) 119894 = 1 2 119896

(ii) 120582119894gt 0 119894 = 1 2 119896 sum119896

119894=1120582119894= 1

such that sum119896

119894=1120582119894119891(120577 120578

119894) + ⟨ℎ(120577) 120583⟩ + ⟨119860119911 119911⟩

12 satisfies thefollowing conditions

119896

sum119894=1

120582119894nabla119911119891 (1205770 120578119894) + nabla119911119891 (1205770 120578119894) + 119860119906

+ (120583119879

nabla119911ℎ (1205770) + 120583119867

nabla119911ℎ (1205770)) = 0

(19)

Re ⟨ℎ (1205770) 120583⟩ = 0 (20)

119906119867

119860119906 le 1 (21)

(119911119867

01198601199110)12

= Re (1199111198670119860119906) (22)

Theorem 14 (sufficient optimality conditions) Let 1205770

=

(1199110 1199110) isin 119876 be a feasible solution to (P) Suppose that there

exists a positive integer 119896 120582119894gt 0 120578119894isin 119884(1205770) 119894 = 1 2 119896with

sum119896

119894=1120582119894= 1 and 0 = 120583 isin 119878lowast sub 119862119901 satisfying conditions (19)ndash

(22) Further if Re[sum119896119894=1

120582119894[119891(120577 120578

119894) + 119911119867119860119906]] is (119865 120572

1 1205881 120579)-

convex with respect to 119877+on 119876 ℎ(120577) is (119865 120572

2 1205882 120579)-convex on

119876with respect to the polyhedral cone 119878 sub 119862119901 and12058811205721(119911 1199110)+

12058821205722(119911 1199110) ge 0 then 120577

0= (1199110 1199110) is an optimal solution to

(P)

Proof We prove this theorem by contradiction Suppose thatthere is a feasible solution 120577 isin 119876 such that

sup120578isin119884

Re [119891 (120577 120578) + (119911119867

119860119911)12

]

lt sup120578isin119884

Re [119891 (1205770 120578) + (119911

119867

01198601199110)12

]

(23)

Since 120578119894isin 119884(1205770) 119894 = 1 2 119896 we have

sup120578isin119884

Re [119891 (1205770 120578) + (119911

119867

01198601199110)12

]

= Re [119891 (1205770 120578119894) + (119911

119867

01198601199110)12

]

for 119894 = 1 2 119896

Re [119891 (120577 120578119894) + (119911

119867

119860119911)12

]

lt sup120578isin119884

Re [119891 (120577 120578) + (119911119867

119860119911)12

]

for 119894 = 1 2 119896

(24)

Thus from the above three inequalities we obtain

Re [119891 (120577 120578119894) + (119911

119867

119860119911)12

]

lt Re [119891 (1205770 120578119894) + (119911

119867

01198601199110)12

]

for 119894 = 1 2 119896

(25)

Journal of Optimization 5

Using (21) and generalized Schwarz inequality we get

Re (119911119867119860119906) le (119911119867

119860119911)12

(119906119867

119860119906)12

le (119911119867

119860119911)12

= Re [(119911119867119860119911)12

]

(26)

and inequality (22) yields

Re (1199111198670119860119906) = Re [(119911119867

01198601199110)12

] (27)

Using (26) and (27) in (25) we have

Re [119891 (120577 120578119894) + 119911119867

119860119906] lt Re [119891 (1205770 120578119894) + 119911119867

0119860119906]

for 119894 = 1 2 119896

(28)

Since 120582119894gt 0 and sum

119896

119894=1120582119894= 1 we have

119903119897Re[119896

sum119894=1

120582119894[119891 (120577 120578

119894) + 119911119867

119860119906] minus

119896

sum119894=1

120582119894[119891 (1205770 120578119894) + 119911119867

0119860119906]]

lt 0

(29)

Since Re[sum119896119894=1

120582119894[119891(120577 120578

119894) + 119911119867119860119906]] is (119865 120572

1 1205881 120579)-convex

with respect to 119877+on 119876 we have

Re[119896

sum119894=1

120582119894[119891 (120577 120578

119894) + 119911119867

119860119906]

minus

119896

sum119894=1

120582119894[119891 (1205770 120578119894) + 119911119867

0119860119906]]

ge 119865[119911 1199110 1205721(119911 1199110)

times(

119896

sum119894=1

120582119894nabla119911119891 (1205770 120578119894) + nabla119911119891 (1205770 120578119894) + 119860119906)]

+ 12058811205792

(119911 1199110)

(30)

From (29) and (30) we conclude that

119865[119911 1199110 1205721(119911 1199110)

times(

119896

sum119894=1

120582119894nabla119911119891(1205770 120578119894) + nabla119911119891 (1205770 120578119894) + 119860119906)]

lt minus12058811205792

(119911 1199110)

(31)

which due to sublinearity of 119865 can be written as

119865[119911 1199110

119896

sum119894=1

120582119894nabla119911119891 (1205770 120578119894) + nabla119911119891 (1205770 120578119894) + 119860119906]

lt minus1205881

1205721(119911 1199110)1205792

(119911 1199110)

(32)

On the other hand from the feasibility of 120577 to (P) we haveminusℎ(120577) isin 119878 or Re⟨ℎ(120577) 120583⟩ le 0 for 120583 isin 119878lowast which along with(20) yields

Re ⟨ℎ (120577) 120583⟩ le 0 = Re ⟨ℎ (1205770) 120583⟩ (33)

Since ℎ(120577) is (119865 1205722 1205882 120579)-convex on 119876 with respect to the

polyhedral cone 119878 sub 119862119901 we have

Re ⟨ℎ (120577) 120583⟩ minus Re ⟨ℎ (1205770) 120583⟩

ge 119865 [119911 1199110 1205722(119911 1199110) (120583119879

nabla119911ℎ (1205770) + 120583119867

nabla119911ℎ (1205770))]

+ 12058821205792

(119911 1199110)

(34)

From (33) and (34) it follows that

119865 [119911 1199110 1205722(119911 1199110) (120583119879

nabla119911ℎ (1205770) + 120583119867

nabla119911ℎ (1205770))]

le minus12058821205792

(119911 1199110)

(35)

which due to sublinearity of 119865 can be written as

119865 [119911 1199110 120583119879

nabla119911ℎ (1205770) + 120583119867

nabla119911ℎ (1205770)] le minus

1205882

1205722(119911 1199110)1205792

(119911 1199110)

(36)

On adding (32) and (36) and using sublinearity of 119865 we get

119865[119911 1199110

119896

sum119894=1

120582119894nabla119911119891 (1205770 120578119894) + nabla119911119891 (1205770 120578119894) + 119860119906

+120583119879

nabla119911ℎ (1205770) + 120583119867

nabla119911ℎ (1205770) ]

lt minus(1205881

1205721(119911 1199110)+

1205882

1205722(119911 1199110)) 1205792

(119911 1199110)

(37)

The above inequality together with the assumption12058811205721(119911 1199110) + 12058821205722(119911 1199110) ge 0 gives

119865[119911 1199110

119896

sum119894=1

120582119894nabla119911119891 (1205770 120578119894) + nabla119911119891 (1205770 120578119894) + 119860119906

+ 120583119879

nabla119911ℎ (1205770) + 120583119867

nabla119911ℎ (1205770) ] lt 0

(38)

which contradicts (19) hence the theorem

6 Journal of Optimization

Theorem 15 (sufficient optimality conditions) Let 1205770

=

(1199110 1199110) isin 119876 be a feasible solution to (P) Suppose that there

exists a positive integer 119896 120582119894gt 0 120578119894isin 119884(1205770) 119894 = 1 2 119896with

sum119896

119894=1120582119894= 1 and 0 = 120583 isin 119878lowast sub 119862119901 satisfying conditions (19)ndash

(22) Further if Re[sum119896119894=1

120582119894[119891(120577 120578

119894) + 119911119867119860119906]] is (119865 120572

1 1205881 120579)-

pseudoconvex with respect to 119877+on 119876 ℎ(120577) is (119865 120572

2 1205882 120579)-

quasiconvex on 119876 with respect to the polyhedral cone 119878 sub 119862119901and 120588

11205721(119911 1199110) + 12058821205722(119911 1199110) ge 0 then 120577

0= (1199110 1199110) is an

optimal solution to (P)

Proof Proceeding as in Theorem 14 we have

Re[119896

sum119894=1

120582119894[119891 (120577 120578

119894) + 119911119867

119860119906] minus

119896

sum119894=1

120582119894[119891 (1205770 120578119894) + 119911119867

0119860119906]]

lt 0

(39)

which by (119865 1205721 1205881 120579)-pseudoconvexity of

Re[sum119896119894=1

120582119894[119891(120577 120578

119894) + 119911119867119860119906]] with respect to 119877

+on 119876

yields

119865[119911 1199110 1205721(119911 1199110)

119896

sum119894=1

120582119894nabla119911119891 (1205770 120578119894) + nabla119911119891 (1205770 120578119894) + 119860119906]

lt minus12058811205792 (119911 119911

0)

(40)

Using the sublinearity of 119865 the above inequality can bewritten as

119865[119911 1199110

119896

sum119894=1

120582119894nabla119911119891 (1205770 120578119894) + nabla119911119891 (1205770 120578119894) + 119860119906]

lt minus1205881

1205721(119911 1199110)1205792 (119911 119911

0)

(41)

On the other hand from the feasibility of 120577 to (P) we haveminusℎ(120577) isin 119878 or Re⟨ℎ(120577) 120583⟩ le 0 for 120583 isin 119878lowast which along with(20) yields

Re ⟨ℎ (120577) 120583⟩ le 0 = Re ⟨ℎ (1205770) 120583⟩ (42)

Since ℎ(120577) is (119865 1205722 1205882 120579)-quasiconvex on 119876 with respect to

the polyhedral cone 119878 sub 119862119901 the above inequality yields

119865 [119911 1199110 1205722(119911 1199110) (120583119879

nabla119911ℎ (1205770) + 120583119867

nabla119911ℎ (1205770))]

le minus12058821205792 (119911 119911

0)

(43)

which due to sublinearity of 119865 can be written as

119865 [119911 1199110 120583119879

nabla119911ℎ (1205770) + 120583

119867

nabla119911ℎ (1205770)] le minus

1205882

1205722(119911 1199110)1205792

(119911 1199110)

(44)

On adding (41) and (44) and using sublinearity of 119865 we get

119865[119911 1199110

119896

sum119894=1

120582119894nabla119911119891 (1205770 120578119894) + nabla119911119891 (1205770 120578119894) + 119860119906

+ 120583119879

nabla119911ℎ (1205770) + 120583119867

nabla119911ℎ (1205770) ]

lt minus(1205881

1205721(119911 1199110)+

1205882

1205722(119911 1199110)) 1205792

(119911 1199110)

(45)

The above inequality together with the assumption12058811205721(119911 1199110) + 12058821205722(119911 1199110) ge 0 gives

119865[119911 1199110

119896

sum119894=1

120582119894nabla119911119891 (1205770 120578119894) + nabla119911119891 (1205770 120578119894) + 119860119906

+120583119879

nabla119911ℎ (1205770) + 120583119867

nabla119911ℎ (1205770) ] lt 0

(46)

which contradicts (19) hence the theorem

Theorem 16 (sufficient optimality conditions) Let 1205770

=

(1199110 1199110) isin 119876 be a feasible solution to (P) Suppose that there

exists a positive integer 119896 120582119894gt 0 120578119894isin 119884(1205770) 119894 = 1 2 119896with

sum119896

119894=1120582119894= 1 and 0 = 120583 isin 119878lowast sub 119862119901 satisfying conditions (19)ndash

(22) Further if Re[sum119896119894=1

120582119894[119891(120577 120578

119894) + 119911119867119860119906]] is (119865 120572

1 1205881 120579)-

quasiconvex with respect to 119877+on119876 ℎ(120577) is strict (119865 120572

2 1205882 120579)-

pseudoconvex on119876 with respect to the polyhedral cone 119878 sub 119862119901

and 12058811205721(119911 1199110) + 12058821205722(119911 1199110) ge 0 then 120577

0= (1199110 1199110) is an

optimal solution to (P)

Proof The proof follows on the similar lines of Theorem 15

4 Parametric Duality

We adopt the following notations in order to simplify theformulation of dual

119870 (120585) = (119896 120578) isin 119873 times 119877119896

+times 1198622119898119896

|

= (1205821 1205822 120582

119896) with

119896

sum119894=1

120582119894= 1

120578 = (1205781 1205782 120578

119896)

with 120578119894isin 119884 (120585) 119894 = 1 2 119896

(47)

for 120585 = (119906 119906) isin 119876 sub 1198622119899Now we formulate a parametric dual problem (D1) with

respect to the complex minimax programming problem (P)as follows

max(119896120582120578)isin119870(120585)

sup(120585120583119908119905)isin119883(119896120582120578)

119905(D1)

Journal of Optimization 7

where119883(119896 120578) denotes the set of all (120585 120583 119908 119905) isin 1198622119899 times119862119901 times

119862119899 times 119877 to satisfy the following conditions119896

sum119894=1

120582119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]

+ 120583119879 nabla119911ℎ (120585) + 120583119867nabla

119911ℎ (120585) = 0

(48)

119896

sum119894=1

120582119894Re119891 (120585 120578

119894) + (119906

119867

119860119906)12

minus 119905 ge 0 (49)

Re ⟨ℎ (120585) 120583⟩ ge 0 (50)

119908119867

119860119908 le 1 (51)

(119906119867

119860119906)12

= Re (119906119867119860119908) (52)

0 = 120583 isin 119878lowast

(53)

If for a triplet (119896 120578) isin 119870(120585) the set 119883(119896 120578) = 0then we define the supremum over 119883(119896 120578) to be minusinfin fornonexecption in the formulation of (D1)

Theorem 17 (weak duality) Let 120577 = (119911 119911) and(119896 120578 120585 120583 119908 119905) be feasible solutions to (P) and (D1)respectively Further if Re[sum119896

119894=1120582119894[119891(120577 120578

119894) + 119911119867119860119908]] is

(119865 1205721 1205881 120579)-pseudoconvex with respect to 119877

+on 119876 ℎ(120577) is

(119865 1205722 1205882 120579)-quasiconvex on 119876 with respect to the polyhedral

cone 119878 sub 119862119901 and 12058811205721(119911 119906) + 120588

21205722(119911 119906) ge 0 Then

sup120578isin119884

Re [119891 (120577 120578) + (119911119867

119860119911)12

] ge 119905 (54)

Proof Suppose on the contrary that

sup120578isin119884

Re [119891 (120577 120578) + (119911119867

119860119911)12

] lt 119905 (55)

By compactness of 119884(120585) sub 119884 in 119862119901 120585 isin (119906 119906) isin 119876 thereexist an integer 119896 gt 0 and finite points 120578

119894isin 119884(120585) 120582

119894gt 0

119894 = 1 2 119896 with sum119896

119894=1120582119894= 1 such that (49) holds From

(49) and (55) we have

Re[119896

sum119894=1

120582119894[119891 (120577 120578

119894) + (119911

119867

119860119911)12

]] lt

119896

sum119894=1

120582119894119905

le

119896

sum119894=1

120582119894Re [119891 (120585 120578

119894) + (119906

119867

119860119906)12

]

(56)

From (51) and the generalized Schwarz inequality we have

Re (119911119867119860119908) le (119911119867

119860119911)12

(119908119867

119860119908)12

le (119911119867

119860119911)12

(57)

Using (52) and (57) in (56) we get

Re[119896

sum119894=1

120582119894[119891 (120577 120578

119894) + 119911119867

119860119908]]

lt Re[119896

sum119894=1

120582119894[119891 (120585 120578

119894) + 119906119867

119860119908]]

(58)

Since Re[sum119896119894=1

120582119894[119891(120577 120578

119894) + 119911119867119860119908]] is (119865 120572

1 1205881 120579)-

pseudoconvex with respect to 119877+on 119876 the above inequality

implies that

119865[119911 119906 1205721(119911 119906)

119896

sum119894=1

120582119894nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]

lt minus12058811205792

(119911 119906)

(59)

which due to sublinearity of 119865 can be written as

119865[119911 119906

119896

sum119894=1

120582119894nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]

lt minus1205881

1205721(119911 119906)

1205792

(119911 119906)

(60)

By the feasibility of 120577 = (119911 119911) to (P) we have minusℎ(120577) isin 119878 orRe⟨ℎ(120577) 120583⟩ le 0 for 120583 isin 119878lowast which along with (50) yields

Re ⟨ℎ (120577) 120583⟩ le 0 le Re ⟨ℎ (120585) 120583⟩ (61)

The above inequality together with the (119865 1205722 1205882 120579)-

quasiconvexity of ℎ(120577) on 119876 with respect to the polyhedralcone 119878 sub 119862

119901 implies

119865 [119911 119906 1205722(119911 119906) (120583

119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

le minus12058821205792

(119911 119906)

(62)

which due to sublinearity of 119865 can be written as

119865 [119911 119906 120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585)]

le minus1205882

1205722(119911 119906)

1205792

(119911 119906) (63)

On adding (60) and (63) and using sublinearity of 119865 we get

119865[119911 119906

119896

sum119894=1

120582119894nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908

+ 120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585) ]

lt minus(1205881

1205721(119911 119906)

+1205882

1205722(119911 119906)

) 1205792

(119911 119906)

(64)

From the assumption 12058811205721(119911 119906) +120588

21205722(119911 119906) ge 0 the above

inequality yields

119865[119911 119906

119896

sum119894=1

120582119894nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908

+ 120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585) ] lt 0

(65)

which contradicts (48) hence the theorem

8 Journal of Optimization

Theorem 18 (weak duality) Let 120577 = (119911 119911) and(119896 120578 120585 120583 119908 119905) be feasible solutions to (P) and (D1)respectively Further if Re[sum119896

119894=1120582119894[119891(120577 120578

119894) + 119911

119867

119860119908]] is(119865 1205721 1205881 120579)-quasiconvex with respect to 119877

+on 119876 ℎ(120577) is

(119865 1205722 1205882 120579)-pseudoconvex on 119876 with respect to the polyhedral

cone 119878 sub 119862119901 and 12058811205721(119911 119906) + 120588

21205722(119911 119906) ge 0 then

sup120578isin119884

Re [119891 (120577 120578) + (119911119867

119860119911)12

] ge 119905 (66)

Proof Theproof follows the same lines as inTheorem 17

Theorem 19 (strong duality) Let 1205770= (1199110 1199110) be an optimal

solution to the problem (P) at which a constraint qualificationis satisfied Then there exist (119896 120578) isin 119870(120577

0) and (120577

0 120583 119908 119905) isin

119883(119896 120578) such that (119896 120578 1205770 120583 119908 119905) is a feasible solution to

the dual problem (D1) If the hypotheses of Theorem 17 or 18are satisfied then (119896 120578 120577

0 120583 119908 119905) is optimal to (D1) and

the two problems (P) and (D1) have the same optimal values

Proof Theproof follows along the lines ofTheorem6 (Lai andLiu [13])

Theorem 20 (strict converse duality) Let 120577 and(

120582 120578 120585 120583 119908 ) be optimal solutions to (P) and (D1)

respectively and assume that the assumptions of Theorem 19are satisfied Further assume that the following conditions aresatisfied

(i) Resum119896

119894=1119894[119891(120577 120578

119894) + 119867119860119908] is strict (119865 120572

1 1205881 120579)-

pseudoconvex with respect to 119877+on 119876 and ℎ(120577) is

(119865 1205722 1205882 120579)-quasiconvex on 119876 with respect to the

polyhedral cone 119878 sub 119862119901(ii) 12058811205721( ) + 120588

21205722( ) ge 0

Then 120577 = 120585 that is 120577 is optimal solution to (D1)

Proof On the contrary suppose that ( ) = 120577 = 120585 = ( )On applyingTheorem 19 we know that

= supisin119884

Re [119891 (120577 120578) + (119867

119860)12

] (67)

From the feasibility of 120577 isin 119876 to (P) 120583 isin 119878lowast and (50) we have

Re ⟨ℎ (120577) 120583⟩ le 0 le Re ⟨ℎ (120585) 120583⟩ (68)

Since ℎ(120577) is (119865 1205722 1205882 120579)-quasiconvex on 119876 with respect to

the polyhedral cone Sin 119862119901 the above inequality yields

119865 [ 1205722( ) (120583

119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

le minus12058821205792

( )

(69)

which by sublinearity of 119865 implies

119865 [ (120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

le minus1205882

1205722( )

1205792

( )

(70)

By (48) and the sublinearity of 119865 we have

119865[

[

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]]

]

+ 119865 [ (120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

ge 119865[

[

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]

+ 120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585)]

]

= 0

(71)

The above inequality together with (70) and 12058811205721( ) +

12058821205722( ) ge 0 gives

119865[

[

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]]

]

ge minus119865 [ (120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

ge1205882

1205722( )

1205792

( )

ge minus1205881

1205721( )

1205792

( )

(72)

That is

119865[

[

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]]

]

ge minus1205881

1205721( )

1205792

( )

(73)

which by sublinearity of 119865 implies

119865[

[

1205721( )

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]]

]

ge minus12058811205792

( )

(74)

Since Resum119896

119894=1119894[119891(120577 120578

119894) + 119867119860119908] is strict (119865 120572

1 1205881 120579)-

pseudoconvex with respect to 119877+on 119876 the above inequality

implies that

Re[

[

119896

sum119894=1

119894[119891 (120577 120578

119894) + 119867

119860119908]]

]

gt Re[

[

119896

sum119894=1

119894[119891 (120585 120578

119894) + 119867

119860119908]]

]

(75)

Journal of Optimization 9

From (51) (52) and the generalized Schwarz inequality wehave

Re (119867119860119908) le (119867

119860)12

Re (119867119860119908) = (119867

119860)12

(76)

which on substituting in (75) and by using (49) we obtain

Re[

[

119896

sum119894=1

119894[119891 (120577 120578

119894) + (

119867

119860)12

]]

]

gt Re[

[

119896

sum119894=1

119894[119891 (120585 120578

119894) + (

119867

119860)12

]]

]

ge

119896

sum119894=1

119894

(77)

Consequently there exist certain 1198940which satisfy

Re [119891 (120577 1205781198940) + (

119867

119860)12

] gt (78)

Hence

supisin119884

Re [119891 (120577 120578) + (119867

119860)12

]

ge Re [119891 (120577 1205781198940) + (

119867

119860)12

] gt

(79)

which contradicts (67) hence the theorem

5 Parameter Free Duality

Making use of the optimality conditions we show thatthe following formation is a dual (D2) to the complexprogramming problem (P)

max(119896120582120578)isin119870(120585)

sup(120585120583119908)isin119883(119896120582120578)

Re [119891 (120585 120578) + (119906119867

119860119906)12

] (D2)

where119883(119896 120578) denotes the set of all (120585 120583 119908) isin 1198622119899times119862119901times119862119899

to satisfy the following conditions

119896

sum119894=1

120582119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]

+ 120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585) = 0

(80)

Re ⟨ℎ (120585) 120583⟩ ge 0 (81)

119908119867

119860119908 le 1 (82)

(119906119867

119860119906)12

= Re (119906119867119860119908) (83)

If for a triplet (119896 120578) isin 119870(120585) the set 119883(119896 120578) = 0then we define the supremum over 119883(119896 120578) to be minusinfin fornonexception in the formulation of (D2)

Nowwe establish appropriate duality theorems and provethat optimal values of (P) and (D2) are equal under theassumption of generalized convexity in order to show that theproblems (P) and (D2) have no duality gap

Theorem21 (weak duality) Let 120577 = (119911 119911) and (119896 120578 120585 120583 119908)

be feasible solutions to (P) and (D2) respectively Furtherif Re[sum119896

119894=1120582119894[119891(120577 120578

119894) + 119911119867119860119908]] is (119865 120572

1 1205881 120579)-pseudoconvex

with respect to 119877+on 119876 ℎ(120577) is (119865 120572

2 1205882 120579)-quasiconvex on 119876

with respect to the polyhedral cone 119878 sub 119862119901 and 12058811205721(z 119906) +

12058821205722(119911 119906) ge 0 then

sup120578isin119884

Re [119891 (120577 120578) + (119911119867

119860119911)12

]

ge sup120578isin119884

Re [119891 (120585 120578) + (119906119867

119860119906)12

]

(84)

Proof On the contrary we suppose that

sup120578isin119884

Re [119891 (120577 120578) + (119911119867

119860119911)12

]

lt sup120578isin119884

Re [119891 (120585 120578) + (119906119867

119860119906)12

]

(85)

Since 120578119894isin 119884(120585) sub 119884 119894 = 1 2 119896 we have

sup120578isin119884

Re [119891 (120585 120578) + (119906119867

119860119906)12

]

= Re [119891 (120585 120578119894) + (119906

119867

119860119906)12

] 119894 = 1 2 119896

(86)

Re [119891 (120577 120578119894) + (119911

119867

119860119911)12

]

le sup120578isin119884

Re [119891 (120577 120578) + (119911119867

119860119911)12

] 119894 = 1 2 119896

(87)

Then the above three inequalities give

Re [119891 (120577 120578119894) + (119911

119867

119860119911)12

]

lt Re [119891 (120585 120578119894) + (119906

119867

119860119906)12

] 119894 = 1 2 119896

(88)

From (82) (83) (88) and the generalized Schwarz inequalitywe have

Re [119891 (120577 120578119894) + (119911

119867

119860119908)]

lt Re [119891 (120585 120578119894) + (119906

119867

119860119908)] 119894 = 1 2 119896

(89)

As 120582119894gt 0 119894 = 1 2 119896 and sum

119896

119894=1120582119894= 1 we have

Re[119896

sum119894=1

120582119894[119891 (120577 120578

119894) + (119911

119867

119860119908)]]

minus Re[119896

sum119894=1

120582119894[119891 (120585 120578

119894) + (119906

119867

119860119908)]] lt 0

(90)

10 Journal of Optimization

Since Re[sum119896119894=1

120582119894[119891(120577 120578

119894) + 119911119867119860119908]] is (119865 120572

1 1205881 120579)-

pseudoconvex with respect to 119877+on 119876 the above inequality

implies that

119865[119911 119906 1205721(119911 119906)

times

119896

sum119894=1

120582119894nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]

lt minus12058811205792

(119911 119906)

(91)

which by sublinearity of 119865 becomes

119865[119911 119906

119896

sum119894=1

120582119894nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]

lt minus1205881

1205721(119911 119906)

1205792

(119911 119906)

(92)

By the feasibility of 120577 = (119911 119911) to (P) 0 = 120583 isin 119878lowast and theinequality (81) we obtain

Re ⟨ℎ (120577) 120583⟩ le 0 le Re ⟨ℎ (120585) 120583⟩ (93)

The above inequality together with the (119865 1205722 1205882 120579)-

quasiconvexity of ℎ(120577) on 119876 with respect to the polyhedralcone 119878 sub 119862

119901 implies

119865 [119911 119906 1205722(119911 119906) (120583

119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

le minus12058821205792

(119911 119906)

(94)

which by sublinearity of 119865 becomes

119865 [119911 119906 120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585)]

le minus1205882

1205722(119911 119906)

1205792

(119911 119906) (95)

On adding (92) and (95) and using the sublinearity of 119865 weget

119865[119911 119906

119896

sum119894=1

120582119894nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908

+120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585) ]

lt minus(1205881

1205721(119911 119906)

+1205882

1205722(119911 119906)

) 1205792

(119911 119906)

(96)

From the assumption 12058811205721(119911 119906) +120588

21205722(119911 119906) ge 0 the above

inequality yields

119865[119911 119906

119896

sum119894=1

120582119894nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908

+ 120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585) ] lt 0

(97)

which contradicts (80) hence the theorem

Theorem 22 (weak duality) Let 120577 = (119911 119911) and(119896 120578 120585 120583 119908) be feasible solutions to (P) and (D2)respectively Further if Re[sum119896

119894=1120582119894[119891(120577 120578

119894) + 119911

119867

119860119908]] is(119865 1205721 1205881 120579)-quasiconvex with respect to 119877

+on 119876 ℎ(120577) is

(119865 1205722 1205882 120579)-pseudoconvex on 119876 with respect to the polyhedral

cone 119878 sub 119862119901 and 12058811205721(119911 119906) + 120588

21205722(119911 119906) ge 0 then

sup120578isin119884

Re [119891 (120577 120578) + (119911119867

119860119911)12

]

ge sup120578isin119884

Re [119891 (120585 120578) + (119906119867

119860119906)12

]

(98)

Proof Theproof follows the same lines as inTheorem 21

Theorem 23 (strong duality) Let 1205770= (1199110 1199110) be an optimal

solution to the problem (P) at which a constraint qualificationis a satisfied Then there exist (119896 120578) isin 119870(120577

0) and (120577

0 120583 119908) isin

119883(119896 120578) such that (119896 120578 1205770 120583 119908) is a feasible solution to the

dual problem (D2) Further if the hypotheses of Theorem 21or Theorem 22 are satisfied then (119896 120578 120577

0 120583 119908) is optimal

to (D2) and the two problems (P) and (D2) have the sameoptimal values

Proof Theproof follows along the lines ofTheorem8 (Lai andLiu [13])

Theorem 24 (strict converse duality) Let 120577 and(

120582 120578 120585 120583 119908) be optimal solutions to (P) and (D2)

respectively and the conditions of Theorem 23 are satisfiedFurther assume that the following conditions are satisfied

(i) Resum119896

119894=1119894[119891(120577 120578

119894) + 119911119867119860119908] is strict (119865 120572

1 1205881 120579)-

pseudoconvex with respect to 119877+on 119876 and ℎ(120577) is

(119865 1205722 1205882 120579)-quasiconvex on 119876 with respect to the

polyhedral cone 119878 sub 119862119901(ii) 12058811205721( ) + 120588

21205722( ) ge 0

Then 120577 = 120585 that is 120585 is an optimal solution to (D2)

Proof On the contrary we assume that ( ) = 120577 = 120585 = ( )On applyingTheorem 23 we know that

supisin119884

Re [119891 (120577 120578) + (119867

119860)12

]

= supisin119884

Re [119891 (120585 120578) + (119867

119860)12

]

(99)

From the feasibility of 120577 isin 119876 to (P) 120583 isin 119878lowast inequality (81)yields

Re ⟨ℎ (120577) 120583⟩ le 0 le Re ⟨ℎ (120585) 120583⟩ (100)

Since ℎ(120577) is (119865 1205722 1205882 120579)-quasiconvex on 119876 with respect to

the polyhedral cone 119878 in 119862119901 the above inequality yields

119865 [ 1205722( ) (120583

119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

le minus12058821205792

( )

(101)

Journal of Optimization 11

which by sublinearity of 119865 implies

119865 [ (120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

le minus1205882

1205722( )

1205792

( )

(102)

By (80) and the sublinearity of 119865 we have

119865[

[

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]]

]

+ 119865 [ (120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

ge 119865[

[

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]

+ 120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585)]

]

= 0

(103)

The above inequality together with (102) and 12058811205721( ) +

12058821205722( ) ge 0 gives

119865[

[

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]]

]

ge minus119865 [ (120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

ge1205882

1205722( )

1205792

( )

ge minus1205881

1205721( )

1205792

( )

(104)

That is

119865[

[

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]]

]

ge minus1205881

1205721( )

1205792 ( )

(105)

which by sublinearity of 119865 implies

119865[

[

1205721( )

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]]

]

ge minus12058811205792 ( )

(106)

Since Resum119896

119894=1119894[119891(120577 120578

119894) + 119911119867119860119908] is strict (119865 120572

1 1205881 120579)-

pseudoconvex with respect to 119877+on 119876 the above inequality

implies that

Re[

[

119896

sum119894=1

119894[119891 (120577 120578

119894) + 119867

119860119908]]

]

gt Re[

[

119896

sum119894=1

119894[119891 (120585 120578

119894) + 119867

119860119908]]

]

(107)

From (82) (83) and the generalized Schwarz inequality weget

Re (119867119860119908) le (119867

119860)12

Re (119867119860119908)12

= (119867

119860)12

(108)

which on substituting in (107) we obtain

Re[

[

119896

sum119894=1

119894[119891 (120577 120578

119894) + (

119867

119860)12

] ]

]

gt Re[

[

119896

sum119894=1

119894[119891 (120585 120578

119894) + (

119867

119860)12

]]

]

(109)

Consequently there exist certain 1198940which satisfy

Re [119891 (120577 1205781198940) + (119867119860)

12

]

gt Re [119891 (120585 1205781198940) + (119867119860)

12

]

(110)

Hence

supisin119884

Re [119891 (120577 120578) + (119867

119860)12

]

ge Re [119891 (120577 1205781198940) + (

119867

119860)12

]

gt Re [119891 (120585 1205781198940) + (

119867

119860)12

]

= supisin119884

Re [119891 (120585 120578) + (119867

119860)12

]

(111)

which contradicts (99) hence the theorem

6 Conclusion

In this paper we introduced generalized (119865 120572 120588 120579)-convexfunctions and established sufficient optimality conditions fora class of nondifferentiable minimax programming problemsin complex space These optimality conditions are thenused to construct two types of dual model and finally wederived weak strong and strict converse duality theorems

12 Journal of Optimization

to show that there is no duality gap between the two dualproblems with respect to the primal problem under somegeneralized convexities of complex functions in the complexprogramming problem As a future task the authors wouldlike to extend these results to second and higher order casesand establish the relations between primal and its second andhigher order dual problems

Acknowledgments

The authors are thankful to the anonymous referees for theirvaluable comments which have improved the presentationof the paper The research of the first author is financiallysupported by the University Grant Commission New DelhiIndia through Grant No F no 41-8012012(SR)

References

[1] N Levinson ldquoLinear programming in complex spacerdquo Journalof Mathematical Analysis and Applications vol 14 no 1 pp 44ndash62 1966

[2] J C Chen and H C Lai ldquoOptimality conditions for minimaxprogramming of analytic functionsrdquoTaiwanese Journal ofMath-ematics vol 8 no 4 pp 673ndash686 2004

[3] N Datta and D Bhatia ldquoDuality for a class of nondifferentiablemathematical programming problems in complex spacerdquo Jour-nal of Mathematical Analysis and Applications vol 101 no 1 pp1ndash11 1984

[4] O P Jain and P C Saxena ldquoA duality theorem for a specialclass of programming problems in complex spacerdquo Journal ofOptimization Theory and Applications vol 16 no 3-4 pp 207ndash220 1975

[5] I M Stancu-Minasian Fractional Programming Theory Meth-ods and Applications vol 409 Kluwer Academic DordrechtGermany 1997

[6] I Ahmad and Z Husain ldquoOptimality conditions and dualityin nondifferentiable minimax fractional programming withgeneralized convexityrdquo Journal of Optimization Theory andApplications vol 129 no 2 pp 255ndash275 2006

[7] A Jayswal I Ahmad and D Kumar ldquoNondifferentiable mini-max fractional programming problem with nonsmooth gener-alized convex functionsrdquoCommunications onAppliedNonlinearAnalysis vol 18 no 4 pp 57ndash75 2011

[8] A Jayswal and D Kumar ldquoOn nondifferentiable minimaxfractional programming involving generalized (C 120572 120588 d)-convexityrdquo Communications on Applied Nonlinear Analysis vol18 no 1 pp 61ndash77 2011

[9] H-C Lai J-C Lee and S-C Ho ldquoParametric duality on mini-max programming involving generalized convexity in complexspacerdquo Journal of Mathematical Analysis and Applications vol323 no 2 pp 1104ndash1115 2006

[10] K Swarup and I C Sharma ldquoProgramming with linear frac-tional functionals in complex spacerdquo Cahiers Center drsquoEtudesRecherche Operationelle vol 12 pp 103ndash109 1970

[11] A Charnes and W W Cooper ldquoProgramming with linearfractional functionalsrdquo Naval Research Logistics Quarterly vol9 no 3-4 pp 181ndash186 1962

[12] H-C Lai and T-Y Huang ldquoOptimality conditions for non-differentiable minimax fractional programming with complexvariablesrdquo Journal of Mathematical Analysis and Applicationsvol 359 no 1 pp 229ndash239 2009

[13] H-C Lai and J-C Liu ldquoDuality for nondifferentiable minimaxprogramming in complex spacesrdquo Nonlinear Analysis TheoryMethods and Applications vol 71 no 12 pp e224ndashe233 2009

[14] I Ahmad Z Husain and S Sharma ldquoSecond-order dualityin nondifferentiable minmax programming involving type-Ifunctionsrdquo Journal of Computational and Applied Mathematicsvol 215 no 1 pp 91ndash102 2008

[15] I Ahmad Z Husain and S Sharma ldquoHigher-order duality innondifferentiableminimax programmingwith generalized typeI functionsrdquo Journal of Optimization Theory and Applicationsvol 141 no 1 pp 1ndash12 2009

[16] BMond andBD Craven ldquoA class of nondifferentiable complexprogramming problemsrdquo Journal of Mathematics and Statisticsvol 6 pp 581ndash591 1975

[17] S K Mishra and N G Rueda ldquoSymmetric duality for math-ematical programming in complex spaces with F-convexityrdquoJournal of Mathematical Analysis and Applications vol 284 no1 pp 250ndash265 2003

[18] H C Lai and J C Liu ldquoMinimax programming in complexspaces-necessary and sufficient optimality conditionsrdquo in Pro-ceedings of the International Symposium on Nonlinear Analysisand Convex Analysis 2007 Research Institute forMathematicalSciences Kyoto University Japan Rims Kokuroku 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Optimization 5

Using (21) and generalized Schwarz inequality we get

Re (119911119867119860119906) le (119911119867

119860119911)12

(119906119867

119860119906)12

le (119911119867

119860119911)12

= Re [(119911119867119860119911)12

]

(26)

and inequality (22) yields

Re (1199111198670119860119906) = Re [(119911119867

01198601199110)12

] (27)

Using (26) and (27) in (25) we have

Re [119891 (120577 120578119894) + 119911119867

119860119906] lt Re [119891 (1205770 120578119894) + 119911119867

0119860119906]

for 119894 = 1 2 119896

(28)

Since 120582119894gt 0 and sum

119896

119894=1120582119894= 1 we have

119903119897Re[119896

sum119894=1

120582119894[119891 (120577 120578

119894) + 119911119867

119860119906] minus

119896

sum119894=1

120582119894[119891 (1205770 120578119894) + 119911119867

0119860119906]]

lt 0

(29)

Since Re[sum119896119894=1

120582119894[119891(120577 120578

119894) + 119911119867119860119906]] is (119865 120572

1 1205881 120579)-convex

with respect to 119877+on 119876 we have

Re[119896

sum119894=1

120582119894[119891 (120577 120578

119894) + 119911119867

119860119906]

minus

119896

sum119894=1

120582119894[119891 (1205770 120578119894) + 119911119867

0119860119906]]

ge 119865[119911 1199110 1205721(119911 1199110)

times(

119896

sum119894=1

120582119894nabla119911119891 (1205770 120578119894) + nabla119911119891 (1205770 120578119894) + 119860119906)]

+ 12058811205792

(119911 1199110)

(30)

From (29) and (30) we conclude that

119865[119911 1199110 1205721(119911 1199110)

times(

119896

sum119894=1

120582119894nabla119911119891(1205770 120578119894) + nabla119911119891 (1205770 120578119894) + 119860119906)]

lt minus12058811205792

(119911 1199110)

(31)

which due to sublinearity of 119865 can be written as

119865[119911 1199110

119896

sum119894=1

120582119894nabla119911119891 (1205770 120578119894) + nabla119911119891 (1205770 120578119894) + 119860119906]

lt minus1205881

1205721(119911 1199110)1205792

(119911 1199110)

(32)

On the other hand from the feasibility of 120577 to (P) we haveminusℎ(120577) isin 119878 or Re⟨ℎ(120577) 120583⟩ le 0 for 120583 isin 119878lowast which along with(20) yields

Re ⟨ℎ (120577) 120583⟩ le 0 = Re ⟨ℎ (1205770) 120583⟩ (33)

Since ℎ(120577) is (119865 1205722 1205882 120579)-convex on 119876 with respect to the

polyhedral cone 119878 sub 119862119901 we have

Re ⟨ℎ (120577) 120583⟩ minus Re ⟨ℎ (1205770) 120583⟩

ge 119865 [119911 1199110 1205722(119911 1199110) (120583119879

nabla119911ℎ (1205770) + 120583119867

nabla119911ℎ (1205770))]

+ 12058821205792

(119911 1199110)

(34)

From (33) and (34) it follows that

119865 [119911 1199110 1205722(119911 1199110) (120583119879

nabla119911ℎ (1205770) + 120583119867

nabla119911ℎ (1205770))]

le minus12058821205792

(119911 1199110)

(35)

which due to sublinearity of 119865 can be written as

119865 [119911 1199110 120583119879

nabla119911ℎ (1205770) + 120583119867

nabla119911ℎ (1205770)] le minus

1205882

1205722(119911 1199110)1205792

(119911 1199110)

(36)

On adding (32) and (36) and using sublinearity of 119865 we get

119865[119911 1199110

119896

sum119894=1

120582119894nabla119911119891 (1205770 120578119894) + nabla119911119891 (1205770 120578119894) + 119860119906

+120583119879

nabla119911ℎ (1205770) + 120583119867

nabla119911ℎ (1205770) ]

lt minus(1205881

1205721(119911 1199110)+

1205882

1205722(119911 1199110)) 1205792

(119911 1199110)

(37)

The above inequality together with the assumption12058811205721(119911 1199110) + 12058821205722(119911 1199110) ge 0 gives

119865[119911 1199110

119896

sum119894=1

120582119894nabla119911119891 (1205770 120578119894) + nabla119911119891 (1205770 120578119894) + 119860119906

+ 120583119879

nabla119911ℎ (1205770) + 120583119867

nabla119911ℎ (1205770) ] lt 0

(38)

which contradicts (19) hence the theorem

6 Journal of Optimization

Theorem 15 (sufficient optimality conditions) Let 1205770

=

(1199110 1199110) isin 119876 be a feasible solution to (P) Suppose that there

exists a positive integer 119896 120582119894gt 0 120578119894isin 119884(1205770) 119894 = 1 2 119896with

sum119896

119894=1120582119894= 1 and 0 = 120583 isin 119878lowast sub 119862119901 satisfying conditions (19)ndash

(22) Further if Re[sum119896119894=1

120582119894[119891(120577 120578

119894) + 119911119867119860119906]] is (119865 120572

1 1205881 120579)-

pseudoconvex with respect to 119877+on 119876 ℎ(120577) is (119865 120572

2 1205882 120579)-

quasiconvex on 119876 with respect to the polyhedral cone 119878 sub 119862119901and 120588

11205721(119911 1199110) + 12058821205722(119911 1199110) ge 0 then 120577

0= (1199110 1199110) is an

optimal solution to (P)

Proof Proceeding as in Theorem 14 we have

Re[119896

sum119894=1

120582119894[119891 (120577 120578

119894) + 119911119867

119860119906] minus

119896

sum119894=1

120582119894[119891 (1205770 120578119894) + 119911119867

0119860119906]]

lt 0

(39)

which by (119865 1205721 1205881 120579)-pseudoconvexity of

Re[sum119896119894=1

120582119894[119891(120577 120578

119894) + 119911119867119860119906]] with respect to 119877

+on 119876

yields

119865[119911 1199110 1205721(119911 1199110)

119896

sum119894=1

120582119894nabla119911119891 (1205770 120578119894) + nabla119911119891 (1205770 120578119894) + 119860119906]

lt minus12058811205792 (119911 119911

0)

(40)

Using the sublinearity of 119865 the above inequality can bewritten as

119865[119911 1199110

119896

sum119894=1

120582119894nabla119911119891 (1205770 120578119894) + nabla119911119891 (1205770 120578119894) + 119860119906]

lt minus1205881

1205721(119911 1199110)1205792 (119911 119911

0)

(41)

On the other hand from the feasibility of 120577 to (P) we haveminusℎ(120577) isin 119878 or Re⟨ℎ(120577) 120583⟩ le 0 for 120583 isin 119878lowast which along with(20) yields

Re ⟨ℎ (120577) 120583⟩ le 0 = Re ⟨ℎ (1205770) 120583⟩ (42)

Since ℎ(120577) is (119865 1205722 1205882 120579)-quasiconvex on 119876 with respect to

the polyhedral cone 119878 sub 119862119901 the above inequality yields

119865 [119911 1199110 1205722(119911 1199110) (120583119879

nabla119911ℎ (1205770) + 120583119867

nabla119911ℎ (1205770))]

le minus12058821205792 (119911 119911

0)

(43)

which due to sublinearity of 119865 can be written as

119865 [119911 1199110 120583119879

nabla119911ℎ (1205770) + 120583

119867

nabla119911ℎ (1205770)] le minus

1205882

1205722(119911 1199110)1205792

(119911 1199110)

(44)

On adding (41) and (44) and using sublinearity of 119865 we get

119865[119911 1199110

119896

sum119894=1

120582119894nabla119911119891 (1205770 120578119894) + nabla119911119891 (1205770 120578119894) + 119860119906

+ 120583119879

nabla119911ℎ (1205770) + 120583119867

nabla119911ℎ (1205770) ]

lt minus(1205881

1205721(119911 1199110)+

1205882

1205722(119911 1199110)) 1205792

(119911 1199110)

(45)

The above inequality together with the assumption12058811205721(119911 1199110) + 12058821205722(119911 1199110) ge 0 gives

119865[119911 1199110

119896

sum119894=1

120582119894nabla119911119891 (1205770 120578119894) + nabla119911119891 (1205770 120578119894) + 119860119906

+120583119879

nabla119911ℎ (1205770) + 120583119867

nabla119911ℎ (1205770) ] lt 0

(46)

which contradicts (19) hence the theorem

Theorem 16 (sufficient optimality conditions) Let 1205770

=

(1199110 1199110) isin 119876 be a feasible solution to (P) Suppose that there

exists a positive integer 119896 120582119894gt 0 120578119894isin 119884(1205770) 119894 = 1 2 119896with

sum119896

119894=1120582119894= 1 and 0 = 120583 isin 119878lowast sub 119862119901 satisfying conditions (19)ndash

(22) Further if Re[sum119896119894=1

120582119894[119891(120577 120578

119894) + 119911119867119860119906]] is (119865 120572

1 1205881 120579)-

quasiconvex with respect to 119877+on119876 ℎ(120577) is strict (119865 120572

2 1205882 120579)-

pseudoconvex on119876 with respect to the polyhedral cone 119878 sub 119862119901

and 12058811205721(119911 1199110) + 12058821205722(119911 1199110) ge 0 then 120577

0= (1199110 1199110) is an

optimal solution to (P)

Proof The proof follows on the similar lines of Theorem 15

4 Parametric Duality

We adopt the following notations in order to simplify theformulation of dual

119870 (120585) = (119896 120578) isin 119873 times 119877119896

+times 1198622119898119896

|

= (1205821 1205822 120582

119896) with

119896

sum119894=1

120582119894= 1

120578 = (1205781 1205782 120578

119896)

with 120578119894isin 119884 (120585) 119894 = 1 2 119896

(47)

for 120585 = (119906 119906) isin 119876 sub 1198622119899Now we formulate a parametric dual problem (D1) with

respect to the complex minimax programming problem (P)as follows

max(119896120582120578)isin119870(120585)

sup(120585120583119908119905)isin119883(119896120582120578)

119905(D1)

Journal of Optimization 7

where119883(119896 120578) denotes the set of all (120585 120583 119908 119905) isin 1198622119899 times119862119901 times

119862119899 times 119877 to satisfy the following conditions119896

sum119894=1

120582119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]

+ 120583119879 nabla119911ℎ (120585) + 120583119867nabla

119911ℎ (120585) = 0

(48)

119896

sum119894=1

120582119894Re119891 (120585 120578

119894) + (119906

119867

119860119906)12

minus 119905 ge 0 (49)

Re ⟨ℎ (120585) 120583⟩ ge 0 (50)

119908119867

119860119908 le 1 (51)

(119906119867

119860119906)12

= Re (119906119867119860119908) (52)

0 = 120583 isin 119878lowast

(53)

If for a triplet (119896 120578) isin 119870(120585) the set 119883(119896 120578) = 0then we define the supremum over 119883(119896 120578) to be minusinfin fornonexecption in the formulation of (D1)

Theorem 17 (weak duality) Let 120577 = (119911 119911) and(119896 120578 120585 120583 119908 119905) be feasible solutions to (P) and (D1)respectively Further if Re[sum119896

119894=1120582119894[119891(120577 120578

119894) + 119911119867119860119908]] is

(119865 1205721 1205881 120579)-pseudoconvex with respect to 119877

+on 119876 ℎ(120577) is

(119865 1205722 1205882 120579)-quasiconvex on 119876 with respect to the polyhedral

cone 119878 sub 119862119901 and 12058811205721(119911 119906) + 120588

21205722(119911 119906) ge 0 Then

sup120578isin119884

Re [119891 (120577 120578) + (119911119867

119860119911)12

] ge 119905 (54)

Proof Suppose on the contrary that

sup120578isin119884

Re [119891 (120577 120578) + (119911119867

119860119911)12

] lt 119905 (55)

By compactness of 119884(120585) sub 119884 in 119862119901 120585 isin (119906 119906) isin 119876 thereexist an integer 119896 gt 0 and finite points 120578

119894isin 119884(120585) 120582

119894gt 0

119894 = 1 2 119896 with sum119896

119894=1120582119894= 1 such that (49) holds From

(49) and (55) we have

Re[119896

sum119894=1

120582119894[119891 (120577 120578

119894) + (119911

119867

119860119911)12

]] lt

119896

sum119894=1

120582119894119905

le

119896

sum119894=1

120582119894Re [119891 (120585 120578

119894) + (119906

119867

119860119906)12

]

(56)

From (51) and the generalized Schwarz inequality we have

Re (119911119867119860119908) le (119911119867

119860119911)12

(119908119867

119860119908)12

le (119911119867

119860119911)12

(57)

Using (52) and (57) in (56) we get

Re[119896

sum119894=1

120582119894[119891 (120577 120578

119894) + 119911119867

119860119908]]

lt Re[119896

sum119894=1

120582119894[119891 (120585 120578

119894) + 119906119867

119860119908]]

(58)

Since Re[sum119896119894=1

120582119894[119891(120577 120578

119894) + 119911119867119860119908]] is (119865 120572

1 1205881 120579)-

pseudoconvex with respect to 119877+on 119876 the above inequality

implies that

119865[119911 119906 1205721(119911 119906)

119896

sum119894=1

120582119894nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]

lt minus12058811205792

(119911 119906)

(59)

which due to sublinearity of 119865 can be written as

119865[119911 119906

119896

sum119894=1

120582119894nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]

lt minus1205881

1205721(119911 119906)

1205792

(119911 119906)

(60)

By the feasibility of 120577 = (119911 119911) to (P) we have minusℎ(120577) isin 119878 orRe⟨ℎ(120577) 120583⟩ le 0 for 120583 isin 119878lowast which along with (50) yields

Re ⟨ℎ (120577) 120583⟩ le 0 le Re ⟨ℎ (120585) 120583⟩ (61)

The above inequality together with the (119865 1205722 1205882 120579)-

quasiconvexity of ℎ(120577) on 119876 with respect to the polyhedralcone 119878 sub 119862

119901 implies

119865 [119911 119906 1205722(119911 119906) (120583

119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

le minus12058821205792

(119911 119906)

(62)

which due to sublinearity of 119865 can be written as

119865 [119911 119906 120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585)]

le minus1205882

1205722(119911 119906)

1205792

(119911 119906) (63)

On adding (60) and (63) and using sublinearity of 119865 we get

119865[119911 119906

119896

sum119894=1

120582119894nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908

+ 120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585) ]

lt minus(1205881

1205721(119911 119906)

+1205882

1205722(119911 119906)

) 1205792

(119911 119906)

(64)

From the assumption 12058811205721(119911 119906) +120588

21205722(119911 119906) ge 0 the above

inequality yields

119865[119911 119906

119896

sum119894=1

120582119894nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908

+ 120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585) ] lt 0

(65)

which contradicts (48) hence the theorem

8 Journal of Optimization

Theorem 18 (weak duality) Let 120577 = (119911 119911) and(119896 120578 120585 120583 119908 119905) be feasible solutions to (P) and (D1)respectively Further if Re[sum119896

119894=1120582119894[119891(120577 120578

119894) + 119911

119867

119860119908]] is(119865 1205721 1205881 120579)-quasiconvex with respect to 119877

+on 119876 ℎ(120577) is

(119865 1205722 1205882 120579)-pseudoconvex on 119876 with respect to the polyhedral

cone 119878 sub 119862119901 and 12058811205721(119911 119906) + 120588

21205722(119911 119906) ge 0 then

sup120578isin119884

Re [119891 (120577 120578) + (119911119867

119860119911)12

] ge 119905 (66)

Proof Theproof follows the same lines as inTheorem 17

Theorem 19 (strong duality) Let 1205770= (1199110 1199110) be an optimal

solution to the problem (P) at which a constraint qualificationis satisfied Then there exist (119896 120578) isin 119870(120577

0) and (120577

0 120583 119908 119905) isin

119883(119896 120578) such that (119896 120578 1205770 120583 119908 119905) is a feasible solution to

the dual problem (D1) If the hypotheses of Theorem 17 or 18are satisfied then (119896 120578 120577

0 120583 119908 119905) is optimal to (D1) and

the two problems (P) and (D1) have the same optimal values

Proof Theproof follows along the lines ofTheorem6 (Lai andLiu [13])

Theorem 20 (strict converse duality) Let 120577 and(

120582 120578 120585 120583 119908 ) be optimal solutions to (P) and (D1)

respectively and assume that the assumptions of Theorem 19are satisfied Further assume that the following conditions aresatisfied

(i) Resum119896

119894=1119894[119891(120577 120578

119894) + 119867119860119908] is strict (119865 120572

1 1205881 120579)-

pseudoconvex with respect to 119877+on 119876 and ℎ(120577) is

(119865 1205722 1205882 120579)-quasiconvex on 119876 with respect to the

polyhedral cone 119878 sub 119862119901(ii) 12058811205721( ) + 120588

21205722( ) ge 0

Then 120577 = 120585 that is 120577 is optimal solution to (D1)

Proof On the contrary suppose that ( ) = 120577 = 120585 = ( )On applyingTheorem 19 we know that

= supisin119884

Re [119891 (120577 120578) + (119867

119860)12

] (67)

From the feasibility of 120577 isin 119876 to (P) 120583 isin 119878lowast and (50) we have

Re ⟨ℎ (120577) 120583⟩ le 0 le Re ⟨ℎ (120585) 120583⟩ (68)

Since ℎ(120577) is (119865 1205722 1205882 120579)-quasiconvex on 119876 with respect to

the polyhedral cone Sin 119862119901 the above inequality yields

119865 [ 1205722( ) (120583

119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

le minus12058821205792

( )

(69)

which by sublinearity of 119865 implies

119865 [ (120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

le minus1205882

1205722( )

1205792

( )

(70)

By (48) and the sublinearity of 119865 we have

119865[

[

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]]

]

+ 119865 [ (120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

ge 119865[

[

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]

+ 120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585)]

]

= 0

(71)

The above inequality together with (70) and 12058811205721( ) +

12058821205722( ) ge 0 gives

119865[

[

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]]

]

ge minus119865 [ (120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

ge1205882

1205722( )

1205792

( )

ge minus1205881

1205721( )

1205792

( )

(72)

That is

119865[

[

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]]

]

ge minus1205881

1205721( )

1205792

( )

(73)

which by sublinearity of 119865 implies

119865[

[

1205721( )

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]]

]

ge minus12058811205792

( )

(74)

Since Resum119896

119894=1119894[119891(120577 120578

119894) + 119867119860119908] is strict (119865 120572

1 1205881 120579)-

pseudoconvex with respect to 119877+on 119876 the above inequality

implies that

Re[

[

119896

sum119894=1

119894[119891 (120577 120578

119894) + 119867

119860119908]]

]

gt Re[

[

119896

sum119894=1

119894[119891 (120585 120578

119894) + 119867

119860119908]]

]

(75)

Journal of Optimization 9

From (51) (52) and the generalized Schwarz inequality wehave

Re (119867119860119908) le (119867

119860)12

Re (119867119860119908) = (119867

119860)12

(76)

which on substituting in (75) and by using (49) we obtain

Re[

[

119896

sum119894=1

119894[119891 (120577 120578

119894) + (

119867

119860)12

]]

]

gt Re[

[

119896

sum119894=1

119894[119891 (120585 120578

119894) + (

119867

119860)12

]]

]

ge

119896

sum119894=1

119894

(77)

Consequently there exist certain 1198940which satisfy

Re [119891 (120577 1205781198940) + (

119867

119860)12

] gt (78)

Hence

supisin119884

Re [119891 (120577 120578) + (119867

119860)12

]

ge Re [119891 (120577 1205781198940) + (

119867

119860)12

] gt

(79)

which contradicts (67) hence the theorem

5 Parameter Free Duality

Making use of the optimality conditions we show thatthe following formation is a dual (D2) to the complexprogramming problem (P)

max(119896120582120578)isin119870(120585)

sup(120585120583119908)isin119883(119896120582120578)

Re [119891 (120585 120578) + (119906119867

119860119906)12

] (D2)

where119883(119896 120578) denotes the set of all (120585 120583 119908) isin 1198622119899times119862119901times119862119899

to satisfy the following conditions

119896

sum119894=1

120582119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]

+ 120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585) = 0

(80)

Re ⟨ℎ (120585) 120583⟩ ge 0 (81)

119908119867

119860119908 le 1 (82)

(119906119867

119860119906)12

= Re (119906119867119860119908) (83)

If for a triplet (119896 120578) isin 119870(120585) the set 119883(119896 120578) = 0then we define the supremum over 119883(119896 120578) to be minusinfin fornonexception in the formulation of (D2)

Nowwe establish appropriate duality theorems and provethat optimal values of (P) and (D2) are equal under theassumption of generalized convexity in order to show that theproblems (P) and (D2) have no duality gap

Theorem21 (weak duality) Let 120577 = (119911 119911) and (119896 120578 120585 120583 119908)

be feasible solutions to (P) and (D2) respectively Furtherif Re[sum119896

119894=1120582119894[119891(120577 120578

119894) + 119911119867119860119908]] is (119865 120572

1 1205881 120579)-pseudoconvex

with respect to 119877+on 119876 ℎ(120577) is (119865 120572

2 1205882 120579)-quasiconvex on 119876

with respect to the polyhedral cone 119878 sub 119862119901 and 12058811205721(z 119906) +

12058821205722(119911 119906) ge 0 then

sup120578isin119884

Re [119891 (120577 120578) + (119911119867

119860119911)12

]

ge sup120578isin119884

Re [119891 (120585 120578) + (119906119867

119860119906)12

]

(84)

Proof On the contrary we suppose that

sup120578isin119884

Re [119891 (120577 120578) + (119911119867

119860119911)12

]

lt sup120578isin119884

Re [119891 (120585 120578) + (119906119867

119860119906)12

]

(85)

Since 120578119894isin 119884(120585) sub 119884 119894 = 1 2 119896 we have

sup120578isin119884

Re [119891 (120585 120578) + (119906119867

119860119906)12

]

= Re [119891 (120585 120578119894) + (119906

119867

119860119906)12

] 119894 = 1 2 119896

(86)

Re [119891 (120577 120578119894) + (119911

119867

119860119911)12

]

le sup120578isin119884

Re [119891 (120577 120578) + (119911119867

119860119911)12

] 119894 = 1 2 119896

(87)

Then the above three inequalities give

Re [119891 (120577 120578119894) + (119911

119867

119860119911)12

]

lt Re [119891 (120585 120578119894) + (119906

119867

119860119906)12

] 119894 = 1 2 119896

(88)

From (82) (83) (88) and the generalized Schwarz inequalitywe have

Re [119891 (120577 120578119894) + (119911

119867

119860119908)]

lt Re [119891 (120585 120578119894) + (119906

119867

119860119908)] 119894 = 1 2 119896

(89)

As 120582119894gt 0 119894 = 1 2 119896 and sum

119896

119894=1120582119894= 1 we have

Re[119896

sum119894=1

120582119894[119891 (120577 120578

119894) + (119911

119867

119860119908)]]

minus Re[119896

sum119894=1

120582119894[119891 (120585 120578

119894) + (119906

119867

119860119908)]] lt 0

(90)

10 Journal of Optimization

Since Re[sum119896119894=1

120582119894[119891(120577 120578

119894) + 119911119867119860119908]] is (119865 120572

1 1205881 120579)-

pseudoconvex with respect to 119877+on 119876 the above inequality

implies that

119865[119911 119906 1205721(119911 119906)

times

119896

sum119894=1

120582119894nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]

lt minus12058811205792

(119911 119906)

(91)

which by sublinearity of 119865 becomes

119865[119911 119906

119896

sum119894=1

120582119894nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]

lt minus1205881

1205721(119911 119906)

1205792

(119911 119906)

(92)

By the feasibility of 120577 = (119911 119911) to (P) 0 = 120583 isin 119878lowast and theinequality (81) we obtain

Re ⟨ℎ (120577) 120583⟩ le 0 le Re ⟨ℎ (120585) 120583⟩ (93)

The above inequality together with the (119865 1205722 1205882 120579)-

quasiconvexity of ℎ(120577) on 119876 with respect to the polyhedralcone 119878 sub 119862

119901 implies

119865 [119911 119906 1205722(119911 119906) (120583

119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

le minus12058821205792

(119911 119906)

(94)

which by sublinearity of 119865 becomes

119865 [119911 119906 120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585)]

le minus1205882

1205722(119911 119906)

1205792

(119911 119906) (95)

On adding (92) and (95) and using the sublinearity of 119865 weget

119865[119911 119906

119896

sum119894=1

120582119894nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908

+120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585) ]

lt minus(1205881

1205721(119911 119906)

+1205882

1205722(119911 119906)

) 1205792

(119911 119906)

(96)

From the assumption 12058811205721(119911 119906) +120588

21205722(119911 119906) ge 0 the above

inequality yields

119865[119911 119906

119896

sum119894=1

120582119894nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908

+ 120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585) ] lt 0

(97)

which contradicts (80) hence the theorem

Theorem 22 (weak duality) Let 120577 = (119911 119911) and(119896 120578 120585 120583 119908) be feasible solutions to (P) and (D2)respectively Further if Re[sum119896

119894=1120582119894[119891(120577 120578

119894) + 119911

119867

119860119908]] is(119865 1205721 1205881 120579)-quasiconvex with respect to 119877

+on 119876 ℎ(120577) is

(119865 1205722 1205882 120579)-pseudoconvex on 119876 with respect to the polyhedral

cone 119878 sub 119862119901 and 12058811205721(119911 119906) + 120588

21205722(119911 119906) ge 0 then

sup120578isin119884

Re [119891 (120577 120578) + (119911119867

119860119911)12

]

ge sup120578isin119884

Re [119891 (120585 120578) + (119906119867

119860119906)12

]

(98)

Proof Theproof follows the same lines as inTheorem 21

Theorem 23 (strong duality) Let 1205770= (1199110 1199110) be an optimal

solution to the problem (P) at which a constraint qualificationis a satisfied Then there exist (119896 120578) isin 119870(120577

0) and (120577

0 120583 119908) isin

119883(119896 120578) such that (119896 120578 1205770 120583 119908) is a feasible solution to the

dual problem (D2) Further if the hypotheses of Theorem 21or Theorem 22 are satisfied then (119896 120578 120577

0 120583 119908) is optimal

to (D2) and the two problems (P) and (D2) have the sameoptimal values

Proof Theproof follows along the lines ofTheorem8 (Lai andLiu [13])

Theorem 24 (strict converse duality) Let 120577 and(

120582 120578 120585 120583 119908) be optimal solutions to (P) and (D2)

respectively and the conditions of Theorem 23 are satisfiedFurther assume that the following conditions are satisfied

(i) Resum119896

119894=1119894[119891(120577 120578

119894) + 119911119867119860119908] is strict (119865 120572

1 1205881 120579)-

pseudoconvex with respect to 119877+on 119876 and ℎ(120577) is

(119865 1205722 1205882 120579)-quasiconvex on 119876 with respect to the

polyhedral cone 119878 sub 119862119901(ii) 12058811205721( ) + 120588

21205722( ) ge 0

Then 120577 = 120585 that is 120585 is an optimal solution to (D2)

Proof On the contrary we assume that ( ) = 120577 = 120585 = ( )On applyingTheorem 23 we know that

supisin119884

Re [119891 (120577 120578) + (119867

119860)12

]

= supisin119884

Re [119891 (120585 120578) + (119867

119860)12

]

(99)

From the feasibility of 120577 isin 119876 to (P) 120583 isin 119878lowast inequality (81)yields

Re ⟨ℎ (120577) 120583⟩ le 0 le Re ⟨ℎ (120585) 120583⟩ (100)

Since ℎ(120577) is (119865 1205722 1205882 120579)-quasiconvex on 119876 with respect to

the polyhedral cone 119878 in 119862119901 the above inequality yields

119865 [ 1205722( ) (120583

119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

le minus12058821205792

( )

(101)

Journal of Optimization 11

which by sublinearity of 119865 implies

119865 [ (120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

le minus1205882

1205722( )

1205792

( )

(102)

By (80) and the sublinearity of 119865 we have

119865[

[

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]]

]

+ 119865 [ (120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

ge 119865[

[

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]

+ 120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585)]

]

= 0

(103)

The above inequality together with (102) and 12058811205721( ) +

12058821205722( ) ge 0 gives

119865[

[

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]]

]

ge minus119865 [ (120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

ge1205882

1205722( )

1205792

( )

ge minus1205881

1205721( )

1205792

( )

(104)

That is

119865[

[

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]]

]

ge minus1205881

1205721( )

1205792 ( )

(105)

which by sublinearity of 119865 implies

119865[

[

1205721( )

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]]

]

ge minus12058811205792 ( )

(106)

Since Resum119896

119894=1119894[119891(120577 120578

119894) + 119911119867119860119908] is strict (119865 120572

1 1205881 120579)-

pseudoconvex with respect to 119877+on 119876 the above inequality

implies that

Re[

[

119896

sum119894=1

119894[119891 (120577 120578

119894) + 119867

119860119908]]

]

gt Re[

[

119896

sum119894=1

119894[119891 (120585 120578

119894) + 119867

119860119908]]

]

(107)

From (82) (83) and the generalized Schwarz inequality weget

Re (119867119860119908) le (119867

119860)12

Re (119867119860119908)12

= (119867

119860)12

(108)

which on substituting in (107) we obtain

Re[

[

119896

sum119894=1

119894[119891 (120577 120578

119894) + (

119867

119860)12

] ]

]

gt Re[

[

119896

sum119894=1

119894[119891 (120585 120578

119894) + (

119867

119860)12

]]

]

(109)

Consequently there exist certain 1198940which satisfy

Re [119891 (120577 1205781198940) + (119867119860)

12

]

gt Re [119891 (120585 1205781198940) + (119867119860)

12

]

(110)

Hence

supisin119884

Re [119891 (120577 120578) + (119867

119860)12

]

ge Re [119891 (120577 1205781198940) + (

119867

119860)12

]

gt Re [119891 (120585 1205781198940) + (

119867

119860)12

]

= supisin119884

Re [119891 (120585 120578) + (119867

119860)12

]

(111)

which contradicts (99) hence the theorem

6 Conclusion

In this paper we introduced generalized (119865 120572 120588 120579)-convexfunctions and established sufficient optimality conditions fora class of nondifferentiable minimax programming problemsin complex space These optimality conditions are thenused to construct two types of dual model and finally wederived weak strong and strict converse duality theorems

12 Journal of Optimization

to show that there is no duality gap between the two dualproblems with respect to the primal problem under somegeneralized convexities of complex functions in the complexprogramming problem As a future task the authors wouldlike to extend these results to second and higher order casesand establish the relations between primal and its second andhigher order dual problems

Acknowledgments

The authors are thankful to the anonymous referees for theirvaluable comments which have improved the presentationof the paper The research of the first author is financiallysupported by the University Grant Commission New DelhiIndia through Grant No F no 41-8012012(SR)

References

[1] N Levinson ldquoLinear programming in complex spacerdquo Journalof Mathematical Analysis and Applications vol 14 no 1 pp 44ndash62 1966

[2] J C Chen and H C Lai ldquoOptimality conditions for minimaxprogramming of analytic functionsrdquoTaiwanese Journal ofMath-ematics vol 8 no 4 pp 673ndash686 2004

[3] N Datta and D Bhatia ldquoDuality for a class of nondifferentiablemathematical programming problems in complex spacerdquo Jour-nal of Mathematical Analysis and Applications vol 101 no 1 pp1ndash11 1984

[4] O P Jain and P C Saxena ldquoA duality theorem for a specialclass of programming problems in complex spacerdquo Journal ofOptimization Theory and Applications vol 16 no 3-4 pp 207ndash220 1975

[5] I M Stancu-Minasian Fractional Programming Theory Meth-ods and Applications vol 409 Kluwer Academic DordrechtGermany 1997

[6] I Ahmad and Z Husain ldquoOptimality conditions and dualityin nondifferentiable minimax fractional programming withgeneralized convexityrdquo Journal of Optimization Theory andApplications vol 129 no 2 pp 255ndash275 2006

[7] A Jayswal I Ahmad and D Kumar ldquoNondifferentiable mini-max fractional programming problem with nonsmooth gener-alized convex functionsrdquoCommunications onAppliedNonlinearAnalysis vol 18 no 4 pp 57ndash75 2011

[8] A Jayswal and D Kumar ldquoOn nondifferentiable minimaxfractional programming involving generalized (C 120572 120588 d)-convexityrdquo Communications on Applied Nonlinear Analysis vol18 no 1 pp 61ndash77 2011

[9] H-C Lai J-C Lee and S-C Ho ldquoParametric duality on mini-max programming involving generalized convexity in complexspacerdquo Journal of Mathematical Analysis and Applications vol323 no 2 pp 1104ndash1115 2006

[10] K Swarup and I C Sharma ldquoProgramming with linear frac-tional functionals in complex spacerdquo Cahiers Center drsquoEtudesRecherche Operationelle vol 12 pp 103ndash109 1970

[11] A Charnes and W W Cooper ldquoProgramming with linearfractional functionalsrdquo Naval Research Logistics Quarterly vol9 no 3-4 pp 181ndash186 1962

[12] H-C Lai and T-Y Huang ldquoOptimality conditions for non-differentiable minimax fractional programming with complexvariablesrdquo Journal of Mathematical Analysis and Applicationsvol 359 no 1 pp 229ndash239 2009

[13] H-C Lai and J-C Liu ldquoDuality for nondifferentiable minimaxprogramming in complex spacesrdquo Nonlinear Analysis TheoryMethods and Applications vol 71 no 12 pp e224ndashe233 2009

[14] I Ahmad Z Husain and S Sharma ldquoSecond-order dualityin nondifferentiable minmax programming involving type-Ifunctionsrdquo Journal of Computational and Applied Mathematicsvol 215 no 1 pp 91ndash102 2008

[15] I Ahmad Z Husain and S Sharma ldquoHigher-order duality innondifferentiableminimax programmingwith generalized typeI functionsrdquo Journal of Optimization Theory and Applicationsvol 141 no 1 pp 1ndash12 2009

[16] BMond andBD Craven ldquoA class of nondifferentiable complexprogramming problemsrdquo Journal of Mathematics and Statisticsvol 6 pp 581ndash591 1975

[17] S K Mishra and N G Rueda ldquoSymmetric duality for math-ematical programming in complex spaces with F-convexityrdquoJournal of Mathematical Analysis and Applications vol 284 no1 pp 250ndash265 2003

[18] H C Lai and J C Liu ldquoMinimax programming in complexspaces-necessary and sufficient optimality conditionsrdquo in Pro-ceedings of the International Symposium on Nonlinear Analysisand Convex Analysis 2007 Research Institute forMathematicalSciences Kyoto University Japan Rims Kokuroku 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Journal of Optimization

Theorem 15 (sufficient optimality conditions) Let 1205770

=

(1199110 1199110) isin 119876 be a feasible solution to (P) Suppose that there

exists a positive integer 119896 120582119894gt 0 120578119894isin 119884(1205770) 119894 = 1 2 119896with

sum119896

119894=1120582119894= 1 and 0 = 120583 isin 119878lowast sub 119862119901 satisfying conditions (19)ndash

(22) Further if Re[sum119896119894=1

120582119894[119891(120577 120578

119894) + 119911119867119860119906]] is (119865 120572

1 1205881 120579)-

pseudoconvex with respect to 119877+on 119876 ℎ(120577) is (119865 120572

2 1205882 120579)-

quasiconvex on 119876 with respect to the polyhedral cone 119878 sub 119862119901and 120588

11205721(119911 1199110) + 12058821205722(119911 1199110) ge 0 then 120577

0= (1199110 1199110) is an

optimal solution to (P)

Proof Proceeding as in Theorem 14 we have

Re[119896

sum119894=1

120582119894[119891 (120577 120578

119894) + 119911119867

119860119906] minus

119896

sum119894=1

120582119894[119891 (1205770 120578119894) + 119911119867

0119860119906]]

lt 0

(39)

which by (119865 1205721 1205881 120579)-pseudoconvexity of

Re[sum119896119894=1

120582119894[119891(120577 120578

119894) + 119911119867119860119906]] with respect to 119877

+on 119876

yields

119865[119911 1199110 1205721(119911 1199110)

119896

sum119894=1

120582119894nabla119911119891 (1205770 120578119894) + nabla119911119891 (1205770 120578119894) + 119860119906]

lt minus12058811205792 (119911 119911

0)

(40)

Using the sublinearity of 119865 the above inequality can bewritten as

119865[119911 1199110

119896

sum119894=1

120582119894nabla119911119891 (1205770 120578119894) + nabla119911119891 (1205770 120578119894) + 119860119906]

lt minus1205881

1205721(119911 1199110)1205792 (119911 119911

0)

(41)

On the other hand from the feasibility of 120577 to (P) we haveminusℎ(120577) isin 119878 or Re⟨ℎ(120577) 120583⟩ le 0 for 120583 isin 119878lowast which along with(20) yields

Re ⟨ℎ (120577) 120583⟩ le 0 = Re ⟨ℎ (1205770) 120583⟩ (42)

Since ℎ(120577) is (119865 1205722 1205882 120579)-quasiconvex on 119876 with respect to

the polyhedral cone 119878 sub 119862119901 the above inequality yields

119865 [119911 1199110 1205722(119911 1199110) (120583119879

nabla119911ℎ (1205770) + 120583119867

nabla119911ℎ (1205770))]

le minus12058821205792 (119911 119911

0)

(43)

which due to sublinearity of 119865 can be written as

119865 [119911 1199110 120583119879

nabla119911ℎ (1205770) + 120583

119867

nabla119911ℎ (1205770)] le minus

1205882

1205722(119911 1199110)1205792

(119911 1199110)

(44)

On adding (41) and (44) and using sublinearity of 119865 we get

119865[119911 1199110

119896

sum119894=1

120582119894nabla119911119891 (1205770 120578119894) + nabla119911119891 (1205770 120578119894) + 119860119906

+ 120583119879

nabla119911ℎ (1205770) + 120583119867

nabla119911ℎ (1205770) ]

lt minus(1205881

1205721(119911 1199110)+

1205882

1205722(119911 1199110)) 1205792

(119911 1199110)

(45)

The above inequality together with the assumption12058811205721(119911 1199110) + 12058821205722(119911 1199110) ge 0 gives

119865[119911 1199110

119896

sum119894=1

120582119894nabla119911119891 (1205770 120578119894) + nabla119911119891 (1205770 120578119894) + 119860119906

+120583119879

nabla119911ℎ (1205770) + 120583119867

nabla119911ℎ (1205770) ] lt 0

(46)

which contradicts (19) hence the theorem

Theorem 16 (sufficient optimality conditions) Let 1205770

=

(1199110 1199110) isin 119876 be a feasible solution to (P) Suppose that there

exists a positive integer 119896 120582119894gt 0 120578119894isin 119884(1205770) 119894 = 1 2 119896with

sum119896

119894=1120582119894= 1 and 0 = 120583 isin 119878lowast sub 119862119901 satisfying conditions (19)ndash

(22) Further if Re[sum119896119894=1

120582119894[119891(120577 120578

119894) + 119911119867119860119906]] is (119865 120572

1 1205881 120579)-

quasiconvex with respect to 119877+on119876 ℎ(120577) is strict (119865 120572

2 1205882 120579)-

pseudoconvex on119876 with respect to the polyhedral cone 119878 sub 119862119901

and 12058811205721(119911 1199110) + 12058821205722(119911 1199110) ge 0 then 120577

0= (1199110 1199110) is an

optimal solution to (P)

Proof The proof follows on the similar lines of Theorem 15

4 Parametric Duality

We adopt the following notations in order to simplify theformulation of dual

119870 (120585) = (119896 120578) isin 119873 times 119877119896

+times 1198622119898119896

|

= (1205821 1205822 120582

119896) with

119896

sum119894=1

120582119894= 1

120578 = (1205781 1205782 120578

119896)

with 120578119894isin 119884 (120585) 119894 = 1 2 119896

(47)

for 120585 = (119906 119906) isin 119876 sub 1198622119899Now we formulate a parametric dual problem (D1) with

respect to the complex minimax programming problem (P)as follows

max(119896120582120578)isin119870(120585)

sup(120585120583119908119905)isin119883(119896120582120578)

119905(D1)

Journal of Optimization 7

where119883(119896 120578) denotes the set of all (120585 120583 119908 119905) isin 1198622119899 times119862119901 times

119862119899 times 119877 to satisfy the following conditions119896

sum119894=1

120582119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]

+ 120583119879 nabla119911ℎ (120585) + 120583119867nabla

119911ℎ (120585) = 0

(48)

119896

sum119894=1

120582119894Re119891 (120585 120578

119894) + (119906

119867

119860119906)12

minus 119905 ge 0 (49)

Re ⟨ℎ (120585) 120583⟩ ge 0 (50)

119908119867

119860119908 le 1 (51)

(119906119867

119860119906)12

= Re (119906119867119860119908) (52)

0 = 120583 isin 119878lowast

(53)

If for a triplet (119896 120578) isin 119870(120585) the set 119883(119896 120578) = 0then we define the supremum over 119883(119896 120578) to be minusinfin fornonexecption in the formulation of (D1)

Theorem 17 (weak duality) Let 120577 = (119911 119911) and(119896 120578 120585 120583 119908 119905) be feasible solutions to (P) and (D1)respectively Further if Re[sum119896

119894=1120582119894[119891(120577 120578

119894) + 119911119867119860119908]] is

(119865 1205721 1205881 120579)-pseudoconvex with respect to 119877

+on 119876 ℎ(120577) is

(119865 1205722 1205882 120579)-quasiconvex on 119876 with respect to the polyhedral

cone 119878 sub 119862119901 and 12058811205721(119911 119906) + 120588

21205722(119911 119906) ge 0 Then

sup120578isin119884

Re [119891 (120577 120578) + (119911119867

119860119911)12

] ge 119905 (54)

Proof Suppose on the contrary that

sup120578isin119884

Re [119891 (120577 120578) + (119911119867

119860119911)12

] lt 119905 (55)

By compactness of 119884(120585) sub 119884 in 119862119901 120585 isin (119906 119906) isin 119876 thereexist an integer 119896 gt 0 and finite points 120578

119894isin 119884(120585) 120582

119894gt 0

119894 = 1 2 119896 with sum119896

119894=1120582119894= 1 such that (49) holds From

(49) and (55) we have

Re[119896

sum119894=1

120582119894[119891 (120577 120578

119894) + (119911

119867

119860119911)12

]] lt

119896

sum119894=1

120582119894119905

le

119896

sum119894=1

120582119894Re [119891 (120585 120578

119894) + (119906

119867

119860119906)12

]

(56)

From (51) and the generalized Schwarz inequality we have

Re (119911119867119860119908) le (119911119867

119860119911)12

(119908119867

119860119908)12

le (119911119867

119860119911)12

(57)

Using (52) and (57) in (56) we get

Re[119896

sum119894=1

120582119894[119891 (120577 120578

119894) + 119911119867

119860119908]]

lt Re[119896

sum119894=1

120582119894[119891 (120585 120578

119894) + 119906119867

119860119908]]

(58)

Since Re[sum119896119894=1

120582119894[119891(120577 120578

119894) + 119911119867119860119908]] is (119865 120572

1 1205881 120579)-

pseudoconvex with respect to 119877+on 119876 the above inequality

implies that

119865[119911 119906 1205721(119911 119906)

119896

sum119894=1

120582119894nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]

lt minus12058811205792

(119911 119906)

(59)

which due to sublinearity of 119865 can be written as

119865[119911 119906

119896

sum119894=1

120582119894nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]

lt minus1205881

1205721(119911 119906)

1205792

(119911 119906)

(60)

By the feasibility of 120577 = (119911 119911) to (P) we have minusℎ(120577) isin 119878 orRe⟨ℎ(120577) 120583⟩ le 0 for 120583 isin 119878lowast which along with (50) yields

Re ⟨ℎ (120577) 120583⟩ le 0 le Re ⟨ℎ (120585) 120583⟩ (61)

The above inequality together with the (119865 1205722 1205882 120579)-

quasiconvexity of ℎ(120577) on 119876 with respect to the polyhedralcone 119878 sub 119862

119901 implies

119865 [119911 119906 1205722(119911 119906) (120583

119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

le minus12058821205792

(119911 119906)

(62)

which due to sublinearity of 119865 can be written as

119865 [119911 119906 120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585)]

le minus1205882

1205722(119911 119906)

1205792

(119911 119906) (63)

On adding (60) and (63) and using sublinearity of 119865 we get

119865[119911 119906

119896

sum119894=1

120582119894nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908

+ 120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585) ]

lt minus(1205881

1205721(119911 119906)

+1205882

1205722(119911 119906)

) 1205792

(119911 119906)

(64)

From the assumption 12058811205721(119911 119906) +120588

21205722(119911 119906) ge 0 the above

inequality yields

119865[119911 119906

119896

sum119894=1

120582119894nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908

+ 120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585) ] lt 0

(65)

which contradicts (48) hence the theorem

8 Journal of Optimization

Theorem 18 (weak duality) Let 120577 = (119911 119911) and(119896 120578 120585 120583 119908 119905) be feasible solutions to (P) and (D1)respectively Further if Re[sum119896

119894=1120582119894[119891(120577 120578

119894) + 119911

119867

119860119908]] is(119865 1205721 1205881 120579)-quasiconvex with respect to 119877

+on 119876 ℎ(120577) is

(119865 1205722 1205882 120579)-pseudoconvex on 119876 with respect to the polyhedral

cone 119878 sub 119862119901 and 12058811205721(119911 119906) + 120588

21205722(119911 119906) ge 0 then

sup120578isin119884

Re [119891 (120577 120578) + (119911119867

119860119911)12

] ge 119905 (66)

Proof Theproof follows the same lines as inTheorem 17

Theorem 19 (strong duality) Let 1205770= (1199110 1199110) be an optimal

solution to the problem (P) at which a constraint qualificationis satisfied Then there exist (119896 120578) isin 119870(120577

0) and (120577

0 120583 119908 119905) isin

119883(119896 120578) such that (119896 120578 1205770 120583 119908 119905) is a feasible solution to

the dual problem (D1) If the hypotheses of Theorem 17 or 18are satisfied then (119896 120578 120577

0 120583 119908 119905) is optimal to (D1) and

the two problems (P) and (D1) have the same optimal values

Proof Theproof follows along the lines ofTheorem6 (Lai andLiu [13])

Theorem 20 (strict converse duality) Let 120577 and(

120582 120578 120585 120583 119908 ) be optimal solutions to (P) and (D1)

respectively and assume that the assumptions of Theorem 19are satisfied Further assume that the following conditions aresatisfied

(i) Resum119896

119894=1119894[119891(120577 120578

119894) + 119867119860119908] is strict (119865 120572

1 1205881 120579)-

pseudoconvex with respect to 119877+on 119876 and ℎ(120577) is

(119865 1205722 1205882 120579)-quasiconvex on 119876 with respect to the

polyhedral cone 119878 sub 119862119901(ii) 12058811205721( ) + 120588

21205722( ) ge 0

Then 120577 = 120585 that is 120577 is optimal solution to (D1)

Proof On the contrary suppose that ( ) = 120577 = 120585 = ( )On applyingTheorem 19 we know that

= supisin119884

Re [119891 (120577 120578) + (119867

119860)12

] (67)

From the feasibility of 120577 isin 119876 to (P) 120583 isin 119878lowast and (50) we have

Re ⟨ℎ (120577) 120583⟩ le 0 le Re ⟨ℎ (120585) 120583⟩ (68)

Since ℎ(120577) is (119865 1205722 1205882 120579)-quasiconvex on 119876 with respect to

the polyhedral cone Sin 119862119901 the above inequality yields

119865 [ 1205722( ) (120583

119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

le minus12058821205792

( )

(69)

which by sublinearity of 119865 implies

119865 [ (120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

le minus1205882

1205722( )

1205792

( )

(70)

By (48) and the sublinearity of 119865 we have

119865[

[

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]]

]

+ 119865 [ (120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

ge 119865[

[

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]

+ 120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585)]

]

= 0

(71)

The above inequality together with (70) and 12058811205721( ) +

12058821205722( ) ge 0 gives

119865[

[

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]]

]

ge minus119865 [ (120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

ge1205882

1205722( )

1205792

( )

ge minus1205881

1205721( )

1205792

( )

(72)

That is

119865[

[

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]]

]

ge minus1205881

1205721( )

1205792

( )

(73)

which by sublinearity of 119865 implies

119865[

[

1205721( )

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]]

]

ge minus12058811205792

( )

(74)

Since Resum119896

119894=1119894[119891(120577 120578

119894) + 119867119860119908] is strict (119865 120572

1 1205881 120579)-

pseudoconvex with respect to 119877+on 119876 the above inequality

implies that

Re[

[

119896

sum119894=1

119894[119891 (120577 120578

119894) + 119867

119860119908]]

]

gt Re[

[

119896

sum119894=1

119894[119891 (120585 120578

119894) + 119867

119860119908]]

]

(75)

Journal of Optimization 9

From (51) (52) and the generalized Schwarz inequality wehave

Re (119867119860119908) le (119867

119860)12

Re (119867119860119908) = (119867

119860)12

(76)

which on substituting in (75) and by using (49) we obtain

Re[

[

119896

sum119894=1

119894[119891 (120577 120578

119894) + (

119867

119860)12

]]

]

gt Re[

[

119896

sum119894=1

119894[119891 (120585 120578

119894) + (

119867

119860)12

]]

]

ge

119896

sum119894=1

119894

(77)

Consequently there exist certain 1198940which satisfy

Re [119891 (120577 1205781198940) + (

119867

119860)12

] gt (78)

Hence

supisin119884

Re [119891 (120577 120578) + (119867

119860)12

]

ge Re [119891 (120577 1205781198940) + (

119867

119860)12

] gt

(79)

which contradicts (67) hence the theorem

5 Parameter Free Duality

Making use of the optimality conditions we show thatthe following formation is a dual (D2) to the complexprogramming problem (P)

max(119896120582120578)isin119870(120585)

sup(120585120583119908)isin119883(119896120582120578)

Re [119891 (120585 120578) + (119906119867

119860119906)12

] (D2)

where119883(119896 120578) denotes the set of all (120585 120583 119908) isin 1198622119899times119862119901times119862119899

to satisfy the following conditions

119896

sum119894=1

120582119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]

+ 120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585) = 0

(80)

Re ⟨ℎ (120585) 120583⟩ ge 0 (81)

119908119867

119860119908 le 1 (82)

(119906119867

119860119906)12

= Re (119906119867119860119908) (83)

If for a triplet (119896 120578) isin 119870(120585) the set 119883(119896 120578) = 0then we define the supremum over 119883(119896 120578) to be minusinfin fornonexception in the formulation of (D2)

Nowwe establish appropriate duality theorems and provethat optimal values of (P) and (D2) are equal under theassumption of generalized convexity in order to show that theproblems (P) and (D2) have no duality gap

Theorem21 (weak duality) Let 120577 = (119911 119911) and (119896 120578 120585 120583 119908)

be feasible solutions to (P) and (D2) respectively Furtherif Re[sum119896

119894=1120582119894[119891(120577 120578

119894) + 119911119867119860119908]] is (119865 120572

1 1205881 120579)-pseudoconvex

with respect to 119877+on 119876 ℎ(120577) is (119865 120572

2 1205882 120579)-quasiconvex on 119876

with respect to the polyhedral cone 119878 sub 119862119901 and 12058811205721(z 119906) +

12058821205722(119911 119906) ge 0 then

sup120578isin119884

Re [119891 (120577 120578) + (119911119867

119860119911)12

]

ge sup120578isin119884

Re [119891 (120585 120578) + (119906119867

119860119906)12

]

(84)

Proof On the contrary we suppose that

sup120578isin119884

Re [119891 (120577 120578) + (119911119867

119860119911)12

]

lt sup120578isin119884

Re [119891 (120585 120578) + (119906119867

119860119906)12

]

(85)

Since 120578119894isin 119884(120585) sub 119884 119894 = 1 2 119896 we have

sup120578isin119884

Re [119891 (120585 120578) + (119906119867

119860119906)12

]

= Re [119891 (120585 120578119894) + (119906

119867

119860119906)12

] 119894 = 1 2 119896

(86)

Re [119891 (120577 120578119894) + (119911

119867

119860119911)12

]

le sup120578isin119884

Re [119891 (120577 120578) + (119911119867

119860119911)12

] 119894 = 1 2 119896

(87)

Then the above three inequalities give

Re [119891 (120577 120578119894) + (119911

119867

119860119911)12

]

lt Re [119891 (120585 120578119894) + (119906

119867

119860119906)12

] 119894 = 1 2 119896

(88)

From (82) (83) (88) and the generalized Schwarz inequalitywe have

Re [119891 (120577 120578119894) + (119911

119867

119860119908)]

lt Re [119891 (120585 120578119894) + (119906

119867

119860119908)] 119894 = 1 2 119896

(89)

As 120582119894gt 0 119894 = 1 2 119896 and sum

119896

119894=1120582119894= 1 we have

Re[119896

sum119894=1

120582119894[119891 (120577 120578

119894) + (119911

119867

119860119908)]]

minus Re[119896

sum119894=1

120582119894[119891 (120585 120578

119894) + (119906

119867

119860119908)]] lt 0

(90)

10 Journal of Optimization

Since Re[sum119896119894=1

120582119894[119891(120577 120578

119894) + 119911119867119860119908]] is (119865 120572

1 1205881 120579)-

pseudoconvex with respect to 119877+on 119876 the above inequality

implies that

119865[119911 119906 1205721(119911 119906)

times

119896

sum119894=1

120582119894nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]

lt minus12058811205792

(119911 119906)

(91)

which by sublinearity of 119865 becomes

119865[119911 119906

119896

sum119894=1

120582119894nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]

lt minus1205881

1205721(119911 119906)

1205792

(119911 119906)

(92)

By the feasibility of 120577 = (119911 119911) to (P) 0 = 120583 isin 119878lowast and theinequality (81) we obtain

Re ⟨ℎ (120577) 120583⟩ le 0 le Re ⟨ℎ (120585) 120583⟩ (93)

The above inequality together with the (119865 1205722 1205882 120579)-

quasiconvexity of ℎ(120577) on 119876 with respect to the polyhedralcone 119878 sub 119862

119901 implies

119865 [119911 119906 1205722(119911 119906) (120583

119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

le minus12058821205792

(119911 119906)

(94)

which by sublinearity of 119865 becomes

119865 [119911 119906 120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585)]

le minus1205882

1205722(119911 119906)

1205792

(119911 119906) (95)

On adding (92) and (95) and using the sublinearity of 119865 weget

119865[119911 119906

119896

sum119894=1

120582119894nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908

+120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585) ]

lt minus(1205881

1205721(119911 119906)

+1205882

1205722(119911 119906)

) 1205792

(119911 119906)

(96)

From the assumption 12058811205721(119911 119906) +120588

21205722(119911 119906) ge 0 the above

inequality yields

119865[119911 119906

119896

sum119894=1

120582119894nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908

+ 120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585) ] lt 0

(97)

which contradicts (80) hence the theorem

Theorem 22 (weak duality) Let 120577 = (119911 119911) and(119896 120578 120585 120583 119908) be feasible solutions to (P) and (D2)respectively Further if Re[sum119896

119894=1120582119894[119891(120577 120578

119894) + 119911

119867

119860119908]] is(119865 1205721 1205881 120579)-quasiconvex with respect to 119877

+on 119876 ℎ(120577) is

(119865 1205722 1205882 120579)-pseudoconvex on 119876 with respect to the polyhedral

cone 119878 sub 119862119901 and 12058811205721(119911 119906) + 120588

21205722(119911 119906) ge 0 then

sup120578isin119884

Re [119891 (120577 120578) + (119911119867

119860119911)12

]

ge sup120578isin119884

Re [119891 (120585 120578) + (119906119867

119860119906)12

]

(98)

Proof Theproof follows the same lines as inTheorem 21

Theorem 23 (strong duality) Let 1205770= (1199110 1199110) be an optimal

solution to the problem (P) at which a constraint qualificationis a satisfied Then there exist (119896 120578) isin 119870(120577

0) and (120577

0 120583 119908) isin

119883(119896 120578) such that (119896 120578 1205770 120583 119908) is a feasible solution to the

dual problem (D2) Further if the hypotheses of Theorem 21or Theorem 22 are satisfied then (119896 120578 120577

0 120583 119908) is optimal

to (D2) and the two problems (P) and (D2) have the sameoptimal values

Proof Theproof follows along the lines ofTheorem8 (Lai andLiu [13])

Theorem 24 (strict converse duality) Let 120577 and(

120582 120578 120585 120583 119908) be optimal solutions to (P) and (D2)

respectively and the conditions of Theorem 23 are satisfiedFurther assume that the following conditions are satisfied

(i) Resum119896

119894=1119894[119891(120577 120578

119894) + 119911119867119860119908] is strict (119865 120572

1 1205881 120579)-

pseudoconvex with respect to 119877+on 119876 and ℎ(120577) is

(119865 1205722 1205882 120579)-quasiconvex on 119876 with respect to the

polyhedral cone 119878 sub 119862119901(ii) 12058811205721( ) + 120588

21205722( ) ge 0

Then 120577 = 120585 that is 120585 is an optimal solution to (D2)

Proof On the contrary we assume that ( ) = 120577 = 120585 = ( )On applyingTheorem 23 we know that

supisin119884

Re [119891 (120577 120578) + (119867

119860)12

]

= supisin119884

Re [119891 (120585 120578) + (119867

119860)12

]

(99)

From the feasibility of 120577 isin 119876 to (P) 120583 isin 119878lowast inequality (81)yields

Re ⟨ℎ (120577) 120583⟩ le 0 le Re ⟨ℎ (120585) 120583⟩ (100)

Since ℎ(120577) is (119865 1205722 1205882 120579)-quasiconvex on 119876 with respect to

the polyhedral cone 119878 in 119862119901 the above inequality yields

119865 [ 1205722( ) (120583

119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

le minus12058821205792

( )

(101)

Journal of Optimization 11

which by sublinearity of 119865 implies

119865 [ (120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

le minus1205882

1205722( )

1205792

( )

(102)

By (80) and the sublinearity of 119865 we have

119865[

[

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]]

]

+ 119865 [ (120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

ge 119865[

[

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]

+ 120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585)]

]

= 0

(103)

The above inequality together with (102) and 12058811205721( ) +

12058821205722( ) ge 0 gives

119865[

[

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]]

]

ge minus119865 [ (120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

ge1205882

1205722( )

1205792

( )

ge minus1205881

1205721( )

1205792

( )

(104)

That is

119865[

[

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]]

]

ge minus1205881

1205721( )

1205792 ( )

(105)

which by sublinearity of 119865 implies

119865[

[

1205721( )

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]]

]

ge minus12058811205792 ( )

(106)

Since Resum119896

119894=1119894[119891(120577 120578

119894) + 119911119867119860119908] is strict (119865 120572

1 1205881 120579)-

pseudoconvex with respect to 119877+on 119876 the above inequality

implies that

Re[

[

119896

sum119894=1

119894[119891 (120577 120578

119894) + 119867

119860119908]]

]

gt Re[

[

119896

sum119894=1

119894[119891 (120585 120578

119894) + 119867

119860119908]]

]

(107)

From (82) (83) and the generalized Schwarz inequality weget

Re (119867119860119908) le (119867

119860)12

Re (119867119860119908)12

= (119867

119860)12

(108)

which on substituting in (107) we obtain

Re[

[

119896

sum119894=1

119894[119891 (120577 120578

119894) + (

119867

119860)12

] ]

]

gt Re[

[

119896

sum119894=1

119894[119891 (120585 120578

119894) + (

119867

119860)12

]]

]

(109)

Consequently there exist certain 1198940which satisfy

Re [119891 (120577 1205781198940) + (119867119860)

12

]

gt Re [119891 (120585 1205781198940) + (119867119860)

12

]

(110)

Hence

supisin119884

Re [119891 (120577 120578) + (119867

119860)12

]

ge Re [119891 (120577 1205781198940) + (

119867

119860)12

]

gt Re [119891 (120585 1205781198940) + (

119867

119860)12

]

= supisin119884

Re [119891 (120585 120578) + (119867

119860)12

]

(111)

which contradicts (99) hence the theorem

6 Conclusion

In this paper we introduced generalized (119865 120572 120588 120579)-convexfunctions and established sufficient optimality conditions fora class of nondifferentiable minimax programming problemsin complex space These optimality conditions are thenused to construct two types of dual model and finally wederived weak strong and strict converse duality theorems

12 Journal of Optimization

to show that there is no duality gap between the two dualproblems with respect to the primal problem under somegeneralized convexities of complex functions in the complexprogramming problem As a future task the authors wouldlike to extend these results to second and higher order casesand establish the relations between primal and its second andhigher order dual problems

Acknowledgments

The authors are thankful to the anonymous referees for theirvaluable comments which have improved the presentationof the paper The research of the first author is financiallysupported by the University Grant Commission New DelhiIndia through Grant No F no 41-8012012(SR)

References

[1] N Levinson ldquoLinear programming in complex spacerdquo Journalof Mathematical Analysis and Applications vol 14 no 1 pp 44ndash62 1966

[2] J C Chen and H C Lai ldquoOptimality conditions for minimaxprogramming of analytic functionsrdquoTaiwanese Journal ofMath-ematics vol 8 no 4 pp 673ndash686 2004

[3] N Datta and D Bhatia ldquoDuality for a class of nondifferentiablemathematical programming problems in complex spacerdquo Jour-nal of Mathematical Analysis and Applications vol 101 no 1 pp1ndash11 1984

[4] O P Jain and P C Saxena ldquoA duality theorem for a specialclass of programming problems in complex spacerdquo Journal ofOptimization Theory and Applications vol 16 no 3-4 pp 207ndash220 1975

[5] I M Stancu-Minasian Fractional Programming Theory Meth-ods and Applications vol 409 Kluwer Academic DordrechtGermany 1997

[6] I Ahmad and Z Husain ldquoOptimality conditions and dualityin nondifferentiable minimax fractional programming withgeneralized convexityrdquo Journal of Optimization Theory andApplications vol 129 no 2 pp 255ndash275 2006

[7] A Jayswal I Ahmad and D Kumar ldquoNondifferentiable mini-max fractional programming problem with nonsmooth gener-alized convex functionsrdquoCommunications onAppliedNonlinearAnalysis vol 18 no 4 pp 57ndash75 2011

[8] A Jayswal and D Kumar ldquoOn nondifferentiable minimaxfractional programming involving generalized (C 120572 120588 d)-convexityrdquo Communications on Applied Nonlinear Analysis vol18 no 1 pp 61ndash77 2011

[9] H-C Lai J-C Lee and S-C Ho ldquoParametric duality on mini-max programming involving generalized convexity in complexspacerdquo Journal of Mathematical Analysis and Applications vol323 no 2 pp 1104ndash1115 2006

[10] K Swarup and I C Sharma ldquoProgramming with linear frac-tional functionals in complex spacerdquo Cahiers Center drsquoEtudesRecherche Operationelle vol 12 pp 103ndash109 1970

[11] A Charnes and W W Cooper ldquoProgramming with linearfractional functionalsrdquo Naval Research Logistics Quarterly vol9 no 3-4 pp 181ndash186 1962

[12] H-C Lai and T-Y Huang ldquoOptimality conditions for non-differentiable minimax fractional programming with complexvariablesrdquo Journal of Mathematical Analysis and Applicationsvol 359 no 1 pp 229ndash239 2009

[13] H-C Lai and J-C Liu ldquoDuality for nondifferentiable minimaxprogramming in complex spacesrdquo Nonlinear Analysis TheoryMethods and Applications vol 71 no 12 pp e224ndashe233 2009

[14] I Ahmad Z Husain and S Sharma ldquoSecond-order dualityin nondifferentiable minmax programming involving type-Ifunctionsrdquo Journal of Computational and Applied Mathematicsvol 215 no 1 pp 91ndash102 2008

[15] I Ahmad Z Husain and S Sharma ldquoHigher-order duality innondifferentiableminimax programmingwith generalized typeI functionsrdquo Journal of Optimization Theory and Applicationsvol 141 no 1 pp 1ndash12 2009

[16] BMond andBD Craven ldquoA class of nondifferentiable complexprogramming problemsrdquo Journal of Mathematics and Statisticsvol 6 pp 581ndash591 1975

[17] S K Mishra and N G Rueda ldquoSymmetric duality for math-ematical programming in complex spaces with F-convexityrdquoJournal of Mathematical Analysis and Applications vol 284 no1 pp 250ndash265 2003

[18] H C Lai and J C Liu ldquoMinimax programming in complexspaces-necessary and sufficient optimality conditionsrdquo in Pro-ceedings of the International Symposium on Nonlinear Analysisand Convex Analysis 2007 Research Institute forMathematicalSciences Kyoto University Japan Rims Kokuroku 2008

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Differential EquationsInternational Journal of

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Stochastic AnalysisInternational Journal of

Journal of Optimization 7

where119883(119896 120578) denotes the set of all (120585 120583 119908 119905) isin 1198622119899 times119862119901 times

119862119899 times 119877 to satisfy the following conditions119896

sum119894=1

120582119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]

+ 120583119879 nabla119911ℎ (120585) + 120583119867nabla

119911ℎ (120585) = 0

(48)

119896

sum119894=1

120582119894Re119891 (120585 120578

119894) + (119906

119867

119860119906)12

minus 119905 ge 0 (49)

Re ⟨ℎ (120585) 120583⟩ ge 0 (50)

119908119867

119860119908 le 1 (51)

(119906119867

119860119906)12

= Re (119906119867119860119908) (52)

0 = 120583 isin 119878lowast

(53)

If for a triplet (119896 120578) isin 119870(120585) the set 119883(119896 120578) = 0then we define the supremum over 119883(119896 120578) to be minusinfin fornonexecption in the formulation of (D1)

Theorem 17 (weak duality) Let 120577 = (119911 119911) and(119896 120578 120585 120583 119908 119905) be feasible solutions to (P) and (D1)respectively Further if Re[sum119896

119894=1120582119894[119891(120577 120578

119894) + 119911119867119860119908]] is

(119865 1205721 1205881 120579)-pseudoconvex with respect to 119877

+on 119876 ℎ(120577) is

(119865 1205722 1205882 120579)-quasiconvex on 119876 with respect to the polyhedral

cone 119878 sub 119862119901 and 12058811205721(119911 119906) + 120588

21205722(119911 119906) ge 0 Then

sup120578isin119884

Re [119891 (120577 120578) + (119911119867

119860119911)12

] ge 119905 (54)

Proof Suppose on the contrary that

sup120578isin119884

Re [119891 (120577 120578) + (119911119867

119860119911)12

] lt 119905 (55)

By compactness of 119884(120585) sub 119884 in 119862119901 120585 isin (119906 119906) isin 119876 thereexist an integer 119896 gt 0 and finite points 120578

119894isin 119884(120585) 120582

119894gt 0

119894 = 1 2 119896 with sum119896

119894=1120582119894= 1 such that (49) holds From

(49) and (55) we have

Re[119896

sum119894=1

120582119894[119891 (120577 120578

119894) + (119911

119867

119860119911)12

]] lt

119896

sum119894=1

120582119894119905

le

119896

sum119894=1

120582119894Re [119891 (120585 120578

119894) + (119906

119867

119860119906)12

]

(56)

From (51) and the generalized Schwarz inequality we have

Re (119911119867119860119908) le (119911119867

119860119911)12

(119908119867

119860119908)12

le (119911119867

119860119911)12

(57)

Using (52) and (57) in (56) we get

Re[119896

sum119894=1

120582119894[119891 (120577 120578

119894) + 119911119867

119860119908]]

lt Re[119896

sum119894=1

120582119894[119891 (120585 120578

119894) + 119906119867

119860119908]]

(58)

Since Re[sum119896119894=1

120582119894[119891(120577 120578

119894) + 119911119867119860119908]] is (119865 120572

1 1205881 120579)-

pseudoconvex with respect to 119877+on 119876 the above inequality

implies that

119865[119911 119906 1205721(119911 119906)

119896

sum119894=1

120582119894nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]

lt minus12058811205792

(119911 119906)

(59)

which due to sublinearity of 119865 can be written as

119865[119911 119906

119896

sum119894=1

120582119894nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]

lt minus1205881

1205721(119911 119906)

1205792

(119911 119906)

(60)

By the feasibility of 120577 = (119911 119911) to (P) we have minusℎ(120577) isin 119878 orRe⟨ℎ(120577) 120583⟩ le 0 for 120583 isin 119878lowast which along with (50) yields

Re ⟨ℎ (120577) 120583⟩ le 0 le Re ⟨ℎ (120585) 120583⟩ (61)

The above inequality together with the (119865 1205722 1205882 120579)-

quasiconvexity of ℎ(120577) on 119876 with respect to the polyhedralcone 119878 sub 119862

119901 implies

119865 [119911 119906 1205722(119911 119906) (120583

119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

le minus12058821205792

(119911 119906)

(62)

which due to sublinearity of 119865 can be written as

119865 [119911 119906 120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585)]

le minus1205882

1205722(119911 119906)

1205792

(119911 119906) (63)

On adding (60) and (63) and using sublinearity of 119865 we get

119865[119911 119906

119896

sum119894=1

120582119894nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908

+ 120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585) ]

lt minus(1205881

1205721(119911 119906)

+1205882

1205722(119911 119906)

) 1205792

(119911 119906)

(64)

From the assumption 12058811205721(119911 119906) +120588

21205722(119911 119906) ge 0 the above

inequality yields

119865[119911 119906

119896

sum119894=1

120582119894nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908

+ 120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585) ] lt 0

(65)

which contradicts (48) hence the theorem

8 Journal of Optimization

Theorem 18 (weak duality) Let 120577 = (119911 119911) and(119896 120578 120585 120583 119908 119905) be feasible solutions to (P) and (D1)respectively Further if Re[sum119896

119894=1120582119894[119891(120577 120578

119894) + 119911

119867

119860119908]] is(119865 1205721 1205881 120579)-quasiconvex with respect to 119877

+on 119876 ℎ(120577) is

(119865 1205722 1205882 120579)-pseudoconvex on 119876 with respect to the polyhedral

cone 119878 sub 119862119901 and 12058811205721(119911 119906) + 120588

21205722(119911 119906) ge 0 then

sup120578isin119884

Re [119891 (120577 120578) + (119911119867

119860119911)12

] ge 119905 (66)

Proof Theproof follows the same lines as inTheorem 17

Theorem 19 (strong duality) Let 1205770= (1199110 1199110) be an optimal

solution to the problem (P) at which a constraint qualificationis satisfied Then there exist (119896 120578) isin 119870(120577

0) and (120577

0 120583 119908 119905) isin

119883(119896 120578) such that (119896 120578 1205770 120583 119908 119905) is a feasible solution to

the dual problem (D1) If the hypotheses of Theorem 17 or 18are satisfied then (119896 120578 120577

0 120583 119908 119905) is optimal to (D1) and

the two problems (P) and (D1) have the same optimal values

Proof Theproof follows along the lines ofTheorem6 (Lai andLiu [13])

Theorem 20 (strict converse duality) Let 120577 and(

120582 120578 120585 120583 119908 ) be optimal solutions to (P) and (D1)

respectively and assume that the assumptions of Theorem 19are satisfied Further assume that the following conditions aresatisfied

(i) Resum119896

119894=1119894[119891(120577 120578

119894) + 119867119860119908] is strict (119865 120572

1 1205881 120579)-

pseudoconvex with respect to 119877+on 119876 and ℎ(120577) is

(119865 1205722 1205882 120579)-quasiconvex on 119876 with respect to the

polyhedral cone 119878 sub 119862119901(ii) 12058811205721( ) + 120588

21205722( ) ge 0

Then 120577 = 120585 that is 120577 is optimal solution to (D1)

Proof On the contrary suppose that ( ) = 120577 = 120585 = ( )On applyingTheorem 19 we know that

= supisin119884

Re [119891 (120577 120578) + (119867

119860)12

] (67)

From the feasibility of 120577 isin 119876 to (P) 120583 isin 119878lowast and (50) we have

Re ⟨ℎ (120577) 120583⟩ le 0 le Re ⟨ℎ (120585) 120583⟩ (68)

Since ℎ(120577) is (119865 1205722 1205882 120579)-quasiconvex on 119876 with respect to

the polyhedral cone Sin 119862119901 the above inequality yields

119865 [ 1205722( ) (120583

119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

le minus12058821205792

( )

(69)

which by sublinearity of 119865 implies

119865 [ (120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

le minus1205882

1205722( )

1205792

( )

(70)

By (48) and the sublinearity of 119865 we have

119865[

[

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]]

]

+ 119865 [ (120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

ge 119865[

[

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]

+ 120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585)]

]

= 0

(71)

The above inequality together with (70) and 12058811205721( ) +

12058821205722( ) ge 0 gives

119865[

[

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]]

]

ge minus119865 [ (120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

ge1205882

1205722( )

1205792

( )

ge minus1205881

1205721( )

1205792

( )

(72)

That is

119865[

[

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]]

]

ge minus1205881

1205721( )

1205792

( )

(73)

which by sublinearity of 119865 implies

119865[

[

1205721( )

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]]

]

ge minus12058811205792

( )

(74)

Since Resum119896

119894=1119894[119891(120577 120578

119894) + 119867119860119908] is strict (119865 120572

1 1205881 120579)-

pseudoconvex with respect to 119877+on 119876 the above inequality

implies that

Re[

[

119896

sum119894=1

119894[119891 (120577 120578

119894) + 119867

119860119908]]

]

gt Re[

[

119896

sum119894=1

119894[119891 (120585 120578

119894) + 119867

119860119908]]

]

(75)

Journal of Optimization 9

From (51) (52) and the generalized Schwarz inequality wehave

Re (119867119860119908) le (119867

119860)12

Re (119867119860119908) = (119867

119860)12

(76)

which on substituting in (75) and by using (49) we obtain

Re[

[

119896

sum119894=1

119894[119891 (120577 120578

119894) + (

119867

119860)12

]]

]

gt Re[

[

119896

sum119894=1

119894[119891 (120585 120578

119894) + (

119867

119860)12

]]

]

ge

119896

sum119894=1

119894

(77)

Consequently there exist certain 1198940which satisfy

Re [119891 (120577 1205781198940) + (

119867

119860)12

] gt (78)

Hence

supisin119884

Re [119891 (120577 120578) + (119867

119860)12

]

ge Re [119891 (120577 1205781198940) + (

119867

119860)12

] gt

(79)

which contradicts (67) hence the theorem

5 Parameter Free Duality

Making use of the optimality conditions we show thatthe following formation is a dual (D2) to the complexprogramming problem (P)

max(119896120582120578)isin119870(120585)

sup(120585120583119908)isin119883(119896120582120578)

Re [119891 (120585 120578) + (119906119867

119860119906)12

] (D2)

where119883(119896 120578) denotes the set of all (120585 120583 119908) isin 1198622119899times119862119901times119862119899

to satisfy the following conditions

119896

sum119894=1

120582119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]

+ 120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585) = 0

(80)

Re ⟨ℎ (120585) 120583⟩ ge 0 (81)

119908119867

119860119908 le 1 (82)

(119906119867

119860119906)12

= Re (119906119867119860119908) (83)

If for a triplet (119896 120578) isin 119870(120585) the set 119883(119896 120578) = 0then we define the supremum over 119883(119896 120578) to be minusinfin fornonexception in the formulation of (D2)

Nowwe establish appropriate duality theorems and provethat optimal values of (P) and (D2) are equal under theassumption of generalized convexity in order to show that theproblems (P) and (D2) have no duality gap

Theorem21 (weak duality) Let 120577 = (119911 119911) and (119896 120578 120585 120583 119908)

be feasible solutions to (P) and (D2) respectively Furtherif Re[sum119896

119894=1120582119894[119891(120577 120578

119894) + 119911119867119860119908]] is (119865 120572

1 1205881 120579)-pseudoconvex

with respect to 119877+on 119876 ℎ(120577) is (119865 120572

2 1205882 120579)-quasiconvex on 119876

with respect to the polyhedral cone 119878 sub 119862119901 and 12058811205721(z 119906) +

12058821205722(119911 119906) ge 0 then

sup120578isin119884

Re [119891 (120577 120578) + (119911119867

119860119911)12

]

ge sup120578isin119884

Re [119891 (120585 120578) + (119906119867

119860119906)12

]

(84)

Proof On the contrary we suppose that

sup120578isin119884

Re [119891 (120577 120578) + (119911119867

119860119911)12

]

lt sup120578isin119884

Re [119891 (120585 120578) + (119906119867

119860119906)12

]

(85)

Since 120578119894isin 119884(120585) sub 119884 119894 = 1 2 119896 we have

sup120578isin119884

Re [119891 (120585 120578) + (119906119867

119860119906)12

]

= Re [119891 (120585 120578119894) + (119906

119867

119860119906)12

] 119894 = 1 2 119896

(86)

Re [119891 (120577 120578119894) + (119911

119867

119860119911)12

]

le sup120578isin119884

Re [119891 (120577 120578) + (119911119867

119860119911)12

] 119894 = 1 2 119896

(87)

Then the above three inequalities give

Re [119891 (120577 120578119894) + (119911

119867

119860119911)12

]

lt Re [119891 (120585 120578119894) + (119906

119867

119860119906)12

] 119894 = 1 2 119896

(88)

From (82) (83) (88) and the generalized Schwarz inequalitywe have

Re [119891 (120577 120578119894) + (119911

119867

119860119908)]

lt Re [119891 (120585 120578119894) + (119906

119867

119860119908)] 119894 = 1 2 119896

(89)

As 120582119894gt 0 119894 = 1 2 119896 and sum

119896

119894=1120582119894= 1 we have

Re[119896

sum119894=1

120582119894[119891 (120577 120578

119894) + (119911

119867

119860119908)]]

minus Re[119896

sum119894=1

120582119894[119891 (120585 120578

119894) + (119906

119867

119860119908)]] lt 0

(90)

10 Journal of Optimization

Since Re[sum119896119894=1

120582119894[119891(120577 120578

119894) + 119911119867119860119908]] is (119865 120572

1 1205881 120579)-

pseudoconvex with respect to 119877+on 119876 the above inequality

implies that

119865[119911 119906 1205721(119911 119906)

times

119896

sum119894=1

120582119894nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]

lt minus12058811205792

(119911 119906)

(91)

which by sublinearity of 119865 becomes

119865[119911 119906

119896

sum119894=1

120582119894nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]

lt minus1205881

1205721(119911 119906)

1205792

(119911 119906)

(92)

By the feasibility of 120577 = (119911 119911) to (P) 0 = 120583 isin 119878lowast and theinequality (81) we obtain

Re ⟨ℎ (120577) 120583⟩ le 0 le Re ⟨ℎ (120585) 120583⟩ (93)

The above inequality together with the (119865 1205722 1205882 120579)-

quasiconvexity of ℎ(120577) on 119876 with respect to the polyhedralcone 119878 sub 119862

119901 implies

119865 [119911 119906 1205722(119911 119906) (120583

119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

le minus12058821205792

(119911 119906)

(94)

which by sublinearity of 119865 becomes

119865 [119911 119906 120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585)]

le minus1205882

1205722(119911 119906)

1205792

(119911 119906) (95)

On adding (92) and (95) and using the sublinearity of 119865 weget

119865[119911 119906

119896

sum119894=1

120582119894nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908

+120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585) ]

lt minus(1205881

1205721(119911 119906)

+1205882

1205722(119911 119906)

) 1205792

(119911 119906)

(96)

From the assumption 12058811205721(119911 119906) +120588

21205722(119911 119906) ge 0 the above

inequality yields

119865[119911 119906

119896

sum119894=1

120582119894nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908

+ 120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585) ] lt 0

(97)

which contradicts (80) hence the theorem

Theorem 22 (weak duality) Let 120577 = (119911 119911) and(119896 120578 120585 120583 119908) be feasible solutions to (P) and (D2)respectively Further if Re[sum119896

119894=1120582119894[119891(120577 120578

119894) + 119911

119867

119860119908]] is(119865 1205721 1205881 120579)-quasiconvex with respect to 119877

+on 119876 ℎ(120577) is

(119865 1205722 1205882 120579)-pseudoconvex on 119876 with respect to the polyhedral

cone 119878 sub 119862119901 and 12058811205721(119911 119906) + 120588

21205722(119911 119906) ge 0 then

sup120578isin119884

Re [119891 (120577 120578) + (119911119867

119860119911)12

]

ge sup120578isin119884

Re [119891 (120585 120578) + (119906119867

119860119906)12

]

(98)

Proof Theproof follows the same lines as inTheorem 21

Theorem 23 (strong duality) Let 1205770= (1199110 1199110) be an optimal

solution to the problem (P) at which a constraint qualificationis a satisfied Then there exist (119896 120578) isin 119870(120577

0) and (120577

0 120583 119908) isin

119883(119896 120578) such that (119896 120578 1205770 120583 119908) is a feasible solution to the

dual problem (D2) Further if the hypotheses of Theorem 21or Theorem 22 are satisfied then (119896 120578 120577

0 120583 119908) is optimal

to (D2) and the two problems (P) and (D2) have the sameoptimal values

Proof Theproof follows along the lines ofTheorem8 (Lai andLiu [13])

Theorem 24 (strict converse duality) Let 120577 and(

120582 120578 120585 120583 119908) be optimal solutions to (P) and (D2)

respectively and the conditions of Theorem 23 are satisfiedFurther assume that the following conditions are satisfied

(i) Resum119896

119894=1119894[119891(120577 120578

119894) + 119911119867119860119908] is strict (119865 120572

1 1205881 120579)-

pseudoconvex with respect to 119877+on 119876 and ℎ(120577) is

(119865 1205722 1205882 120579)-quasiconvex on 119876 with respect to the

polyhedral cone 119878 sub 119862119901(ii) 12058811205721( ) + 120588

21205722( ) ge 0

Then 120577 = 120585 that is 120585 is an optimal solution to (D2)

Proof On the contrary we assume that ( ) = 120577 = 120585 = ( )On applyingTheorem 23 we know that

supisin119884

Re [119891 (120577 120578) + (119867

119860)12

]

= supisin119884

Re [119891 (120585 120578) + (119867

119860)12

]

(99)

From the feasibility of 120577 isin 119876 to (P) 120583 isin 119878lowast inequality (81)yields

Re ⟨ℎ (120577) 120583⟩ le 0 le Re ⟨ℎ (120585) 120583⟩ (100)

Since ℎ(120577) is (119865 1205722 1205882 120579)-quasiconvex on 119876 with respect to

the polyhedral cone 119878 in 119862119901 the above inequality yields

119865 [ 1205722( ) (120583

119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

le minus12058821205792

( )

(101)

Journal of Optimization 11

which by sublinearity of 119865 implies

119865 [ (120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

le minus1205882

1205722( )

1205792

( )

(102)

By (80) and the sublinearity of 119865 we have

119865[

[

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]]

]

+ 119865 [ (120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

ge 119865[

[

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]

+ 120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585)]

]

= 0

(103)

The above inequality together with (102) and 12058811205721( ) +

12058821205722( ) ge 0 gives

119865[

[

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]]

]

ge minus119865 [ (120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

ge1205882

1205722( )

1205792

( )

ge minus1205881

1205721( )

1205792

( )

(104)

That is

119865[

[

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]]

]

ge minus1205881

1205721( )

1205792 ( )

(105)

which by sublinearity of 119865 implies

119865[

[

1205721( )

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]]

]

ge minus12058811205792 ( )

(106)

Since Resum119896

119894=1119894[119891(120577 120578

119894) + 119911119867119860119908] is strict (119865 120572

1 1205881 120579)-

pseudoconvex with respect to 119877+on 119876 the above inequality

implies that

Re[

[

119896

sum119894=1

119894[119891 (120577 120578

119894) + 119867

119860119908]]

]

gt Re[

[

119896

sum119894=1

119894[119891 (120585 120578

119894) + 119867

119860119908]]

]

(107)

From (82) (83) and the generalized Schwarz inequality weget

Re (119867119860119908) le (119867

119860)12

Re (119867119860119908)12

= (119867

119860)12

(108)

which on substituting in (107) we obtain

Re[

[

119896

sum119894=1

119894[119891 (120577 120578

119894) + (

119867

119860)12

] ]

]

gt Re[

[

119896

sum119894=1

119894[119891 (120585 120578

119894) + (

119867

119860)12

]]

]

(109)

Consequently there exist certain 1198940which satisfy

Re [119891 (120577 1205781198940) + (119867119860)

12

]

gt Re [119891 (120585 1205781198940) + (119867119860)

12

]

(110)

Hence

supisin119884

Re [119891 (120577 120578) + (119867

119860)12

]

ge Re [119891 (120577 1205781198940) + (

119867

119860)12

]

gt Re [119891 (120585 1205781198940) + (

119867

119860)12

]

= supisin119884

Re [119891 (120585 120578) + (119867

119860)12

]

(111)

which contradicts (99) hence the theorem

6 Conclusion

In this paper we introduced generalized (119865 120572 120588 120579)-convexfunctions and established sufficient optimality conditions fora class of nondifferentiable minimax programming problemsin complex space These optimality conditions are thenused to construct two types of dual model and finally wederived weak strong and strict converse duality theorems

12 Journal of Optimization

to show that there is no duality gap between the two dualproblems with respect to the primal problem under somegeneralized convexities of complex functions in the complexprogramming problem As a future task the authors wouldlike to extend these results to second and higher order casesand establish the relations between primal and its second andhigher order dual problems

Acknowledgments

The authors are thankful to the anonymous referees for theirvaluable comments which have improved the presentationof the paper The research of the first author is financiallysupported by the University Grant Commission New DelhiIndia through Grant No F no 41-8012012(SR)

References

[1] N Levinson ldquoLinear programming in complex spacerdquo Journalof Mathematical Analysis and Applications vol 14 no 1 pp 44ndash62 1966

[2] J C Chen and H C Lai ldquoOptimality conditions for minimaxprogramming of analytic functionsrdquoTaiwanese Journal ofMath-ematics vol 8 no 4 pp 673ndash686 2004

[3] N Datta and D Bhatia ldquoDuality for a class of nondifferentiablemathematical programming problems in complex spacerdquo Jour-nal of Mathematical Analysis and Applications vol 101 no 1 pp1ndash11 1984

[4] O P Jain and P C Saxena ldquoA duality theorem for a specialclass of programming problems in complex spacerdquo Journal ofOptimization Theory and Applications vol 16 no 3-4 pp 207ndash220 1975

[5] I M Stancu-Minasian Fractional Programming Theory Meth-ods and Applications vol 409 Kluwer Academic DordrechtGermany 1997

[6] I Ahmad and Z Husain ldquoOptimality conditions and dualityin nondifferentiable minimax fractional programming withgeneralized convexityrdquo Journal of Optimization Theory andApplications vol 129 no 2 pp 255ndash275 2006

[7] A Jayswal I Ahmad and D Kumar ldquoNondifferentiable mini-max fractional programming problem with nonsmooth gener-alized convex functionsrdquoCommunications onAppliedNonlinearAnalysis vol 18 no 4 pp 57ndash75 2011

[8] A Jayswal and D Kumar ldquoOn nondifferentiable minimaxfractional programming involving generalized (C 120572 120588 d)-convexityrdquo Communications on Applied Nonlinear Analysis vol18 no 1 pp 61ndash77 2011

[9] H-C Lai J-C Lee and S-C Ho ldquoParametric duality on mini-max programming involving generalized convexity in complexspacerdquo Journal of Mathematical Analysis and Applications vol323 no 2 pp 1104ndash1115 2006

[10] K Swarup and I C Sharma ldquoProgramming with linear frac-tional functionals in complex spacerdquo Cahiers Center drsquoEtudesRecherche Operationelle vol 12 pp 103ndash109 1970

[11] A Charnes and W W Cooper ldquoProgramming with linearfractional functionalsrdquo Naval Research Logistics Quarterly vol9 no 3-4 pp 181ndash186 1962

[12] H-C Lai and T-Y Huang ldquoOptimality conditions for non-differentiable minimax fractional programming with complexvariablesrdquo Journal of Mathematical Analysis and Applicationsvol 359 no 1 pp 229ndash239 2009

[13] H-C Lai and J-C Liu ldquoDuality for nondifferentiable minimaxprogramming in complex spacesrdquo Nonlinear Analysis TheoryMethods and Applications vol 71 no 12 pp e224ndashe233 2009

[14] I Ahmad Z Husain and S Sharma ldquoSecond-order dualityin nondifferentiable minmax programming involving type-Ifunctionsrdquo Journal of Computational and Applied Mathematicsvol 215 no 1 pp 91ndash102 2008

[15] I Ahmad Z Husain and S Sharma ldquoHigher-order duality innondifferentiableminimax programmingwith generalized typeI functionsrdquo Journal of Optimization Theory and Applicationsvol 141 no 1 pp 1ndash12 2009

[16] BMond andBD Craven ldquoA class of nondifferentiable complexprogramming problemsrdquo Journal of Mathematics and Statisticsvol 6 pp 581ndash591 1975

[17] S K Mishra and N G Rueda ldquoSymmetric duality for math-ematical programming in complex spaces with F-convexityrdquoJournal of Mathematical Analysis and Applications vol 284 no1 pp 250ndash265 2003

[18] H C Lai and J C Liu ldquoMinimax programming in complexspaces-necessary and sufficient optimality conditionsrdquo in Pro-ceedings of the International Symposium on Nonlinear Analysisand Convex Analysis 2007 Research Institute forMathematicalSciences Kyoto University Japan Rims Kokuroku 2008

Submit your manuscripts athttpwwwhindawicom

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Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Journal of Optimization

Theorem 18 (weak duality) Let 120577 = (119911 119911) and(119896 120578 120585 120583 119908 119905) be feasible solutions to (P) and (D1)respectively Further if Re[sum119896

119894=1120582119894[119891(120577 120578

119894) + 119911

119867

119860119908]] is(119865 1205721 1205881 120579)-quasiconvex with respect to 119877

+on 119876 ℎ(120577) is

(119865 1205722 1205882 120579)-pseudoconvex on 119876 with respect to the polyhedral

cone 119878 sub 119862119901 and 12058811205721(119911 119906) + 120588

21205722(119911 119906) ge 0 then

sup120578isin119884

Re [119891 (120577 120578) + (119911119867

119860119911)12

] ge 119905 (66)

Proof Theproof follows the same lines as inTheorem 17

Theorem 19 (strong duality) Let 1205770= (1199110 1199110) be an optimal

solution to the problem (P) at which a constraint qualificationis satisfied Then there exist (119896 120578) isin 119870(120577

0) and (120577

0 120583 119908 119905) isin

119883(119896 120578) such that (119896 120578 1205770 120583 119908 119905) is a feasible solution to

the dual problem (D1) If the hypotheses of Theorem 17 or 18are satisfied then (119896 120578 120577

0 120583 119908 119905) is optimal to (D1) and

the two problems (P) and (D1) have the same optimal values

Proof Theproof follows along the lines ofTheorem6 (Lai andLiu [13])

Theorem 20 (strict converse duality) Let 120577 and(

120582 120578 120585 120583 119908 ) be optimal solutions to (P) and (D1)

respectively and assume that the assumptions of Theorem 19are satisfied Further assume that the following conditions aresatisfied

(i) Resum119896

119894=1119894[119891(120577 120578

119894) + 119867119860119908] is strict (119865 120572

1 1205881 120579)-

pseudoconvex with respect to 119877+on 119876 and ℎ(120577) is

(119865 1205722 1205882 120579)-quasiconvex on 119876 with respect to the

polyhedral cone 119878 sub 119862119901(ii) 12058811205721( ) + 120588

21205722( ) ge 0

Then 120577 = 120585 that is 120577 is optimal solution to (D1)

Proof On the contrary suppose that ( ) = 120577 = 120585 = ( )On applyingTheorem 19 we know that

= supisin119884

Re [119891 (120577 120578) + (119867

119860)12

] (67)

From the feasibility of 120577 isin 119876 to (P) 120583 isin 119878lowast and (50) we have

Re ⟨ℎ (120577) 120583⟩ le 0 le Re ⟨ℎ (120585) 120583⟩ (68)

Since ℎ(120577) is (119865 1205722 1205882 120579)-quasiconvex on 119876 with respect to

the polyhedral cone Sin 119862119901 the above inequality yields

119865 [ 1205722( ) (120583

119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

le minus12058821205792

( )

(69)

which by sublinearity of 119865 implies

119865 [ (120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

le minus1205882

1205722( )

1205792

( )

(70)

By (48) and the sublinearity of 119865 we have

119865[

[

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]]

]

+ 119865 [ (120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

ge 119865[

[

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]

+ 120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585)]

]

= 0

(71)

The above inequality together with (70) and 12058811205721( ) +

12058821205722( ) ge 0 gives

119865[

[

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]]

]

ge minus119865 [ (120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

ge1205882

1205722( )

1205792

( )

ge minus1205881

1205721( )

1205792

( )

(72)

That is

119865[

[

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]]

]

ge minus1205881

1205721( )

1205792

( )

(73)

which by sublinearity of 119865 implies

119865[

[

1205721( )

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]]

]

ge minus12058811205792

( )

(74)

Since Resum119896

119894=1119894[119891(120577 120578

119894) + 119867119860119908] is strict (119865 120572

1 1205881 120579)-

pseudoconvex with respect to 119877+on 119876 the above inequality

implies that

Re[

[

119896

sum119894=1

119894[119891 (120577 120578

119894) + 119867

119860119908]]

]

gt Re[

[

119896

sum119894=1

119894[119891 (120585 120578

119894) + 119867

119860119908]]

]

(75)

Journal of Optimization 9

From (51) (52) and the generalized Schwarz inequality wehave

Re (119867119860119908) le (119867

119860)12

Re (119867119860119908) = (119867

119860)12

(76)

which on substituting in (75) and by using (49) we obtain

Re[

[

119896

sum119894=1

119894[119891 (120577 120578

119894) + (

119867

119860)12

]]

]

gt Re[

[

119896

sum119894=1

119894[119891 (120585 120578

119894) + (

119867

119860)12

]]

]

ge

119896

sum119894=1

119894

(77)

Consequently there exist certain 1198940which satisfy

Re [119891 (120577 1205781198940) + (

119867

119860)12

] gt (78)

Hence

supisin119884

Re [119891 (120577 120578) + (119867

119860)12

]

ge Re [119891 (120577 1205781198940) + (

119867

119860)12

] gt

(79)

which contradicts (67) hence the theorem

5 Parameter Free Duality

Making use of the optimality conditions we show thatthe following formation is a dual (D2) to the complexprogramming problem (P)

max(119896120582120578)isin119870(120585)

sup(120585120583119908)isin119883(119896120582120578)

Re [119891 (120585 120578) + (119906119867

119860119906)12

] (D2)

where119883(119896 120578) denotes the set of all (120585 120583 119908) isin 1198622119899times119862119901times119862119899

to satisfy the following conditions

119896

sum119894=1

120582119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]

+ 120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585) = 0

(80)

Re ⟨ℎ (120585) 120583⟩ ge 0 (81)

119908119867

119860119908 le 1 (82)

(119906119867

119860119906)12

= Re (119906119867119860119908) (83)

If for a triplet (119896 120578) isin 119870(120585) the set 119883(119896 120578) = 0then we define the supremum over 119883(119896 120578) to be minusinfin fornonexception in the formulation of (D2)

Nowwe establish appropriate duality theorems and provethat optimal values of (P) and (D2) are equal under theassumption of generalized convexity in order to show that theproblems (P) and (D2) have no duality gap

Theorem21 (weak duality) Let 120577 = (119911 119911) and (119896 120578 120585 120583 119908)

be feasible solutions to (P) and (D2) respectively Furtherif Re[sum119896

119894=1120582119894[119891(120577 120578

119894) + 119911119867119860119908]] is (119865 120572

1 1205881 120579)-pseudoconvex

with respect to 119877+on 119876 ℎ(120577) is (119865 120572

2 1205882 120579)-quasiconvex on 119876

with respect to the polyhedral cone 119878 sub 119862119901 and 12058811205721(z 119906) +

12058821205722(119911 119906) ge 0 then

sup120578isin119884

Re [119891 (120577 120578) + (119911119867

119860119911)12

]

ge sup120578isin119884

Re [119891 (120585 120578) + (119906119867

119860119906)12

]

(84)

Proof On the contrary we suppose that

sup120578isin119884

Re [119891 (120577 120578) + (119911119867

119860119911)12

]

lt sup120578isin119884

Re [119891 (120585 120578) + (119906119867

119860119906)12

]

(85)

Since 120578119894isin 119884(120585) sub 119884 119894 = 1 2 119896 we have

sup120578isin119884

Re [119891 (120585 120578) + (119906119867

119860119906)12

]

= Re [119891 (120585 120578119894) + (119906

119867

119860119906)12

] 119894 = 1 2 119896

(86)

Re [119891 (120577 120578119894) + (119911

119867

119860119911)12

]

le sup120578isin119884

Re [119891 (120577 120578) + (119911119867

119860119911)12

] 119894 = 1 2 119896

(87)

Then the above three inequalities give

Re [119891 (120577 120578119894) + (119911

119867

119860119911)12

]

lt Re [119891 (120585 120578119894) + (119906

119867

119860119906)12

] 119894 = 1 2 119896

(88)

From (82) (83) (88) and the generalized Schwarz inequalitywe have

Re [119891 (120577 120578119894) + (119911

119867

119860119908)]

lt Re [119891 (120585 120578119894) + (119906

119867

119860119908)] 119894 = 1 2 119896

(89)

As 120582119894gt 0 119894 = 1 2 119896 and sum

119896

119894=1120582119894= 1 we have

Re[119896

sum119894=1

120582119894[119891 (120577 120578

119894) + (119911

119867

119860119908)]]

minus Re[119896

sum119894=1

120582119894[119891 (120585 120578

119894) + (119906

119867

119860119908)]] lt 0

(90)

10 Journal of Optimization

Since Re[sum119896119894=1

120582119894[119891(120577 120578

119894) + 119911119867119860119908]] is (119865 120572

1 1205881 120579)-

pseudoconvex with respect to 119877+on 119876 the above inequality

implies that

119865[119911 119906 1205721(119911 119906)

times

119896

sum119894=1

120582119894nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]

lt minus12058811205792

(119911 119906)

(91)

which by sublinearity of 119865 becomes

119865[119911 119906

119896

sum119894=1

120582119894nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]

lt minus1205881

1205721(119911 119906)

1205792

(119911 119906)

(92)

By the feasibility of 120577 = (119911 119911) to (P) 0 = 120583 isin 119878lowast and theinequality (81) we obtain

Re ⟨ℎ (120577) 120583⟩ le 0 le Re ⟨ℎ (120585) 120583⟩ (93)

The above inequality together with the (119865 1205722 1205882 120579)-

quasiconvexity of ℎ(120577) on 119876 with respect to the polyhedralcone 119878 sub 119862

119901 implies

119865 [119911 119906 1205722(119911 119906) (120583

119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

le minus12058821205792

(119911 119906)

(94)

which by sublinearity of 119865 becomes

119865 [119911 119906 120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585)]

le minus1205882

1205722(119911 119906)

1205792

(119911 119906) (95)

On adding (92) and (95) and using the sublinearity of 119865 weget

119865[119911 119906

119896

sum119894=1

120582119894nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908

+120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585) ]

lt minus(1205881

1205721(119911 119906)

+1205882

1205722(119911 119906)

) 1205792

(119911 119906)

(96)

From the assumption 12058811205721(119911 119906) +120588

21205722(119911 119906) ge 0 the above

inequality yields

119865[119911 119906

119896

sum119894=1

120582119894nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908

+ 120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585) ] lt 0

(97)

which contradicts (80) hence the theorem

Theorem 22 (weak duality) Let 120577 = (119911 119911) and(119896 120578 120585 120583 119908) be feasible solutions to (P) and (D2)respectively Further if Re[sum119896

119894=1120582119894[119891(120577 120578

119894) + 119911

119867

119860119908]] is(119865 1205721 1205881 120579)-quasiconvex with respect to 119877

+on 119876 ℎ(120577) is

(119865 1205722 1205882 120579)-pseudoconvex on 119876 with respect to the polyhedral

cone 119878 sub 119862119901 and 12058811205721(119911 119906) + 120588

21205722(119911 119906) ge 0 then

sup120578isin119884

Re [119891 (120577 120578) + (119911119867

119860119911)12

]

ge sup120578isin119884

Re [119891 (120585 120578) + (119906119867

119860119906)12

]

(98)

Proof Theproof follows the same lines as inTheorem 21

Theorem 23 (strong duality) Let 1205770= (1199110 1199110) be an optimal

solution to the problem (P) at which a constraint qualificationis a satisfied Then there exist (119896 120578) isin 119870(120577

0) and (120577

0 120583 119908) isin

119883(119896 120578) such that (119896 120578 1205770 120583 119908) is a feasible solution to the

dual problem (D2) Further if the hypotheses of Theorem 21or Theorem 22 are satisfied then (119896 120578 120577

0 120583 119908) is optimal

to (D2) and the two problems (P) and (D2) have the sameoptimal values

Proof Theproof follows along the lines ofTheorem8 (Lai andLiu [13])

Theorem 24 (strict converse duality) Let 120577 and(

120582 120578 120585 120583 119908) be optimal solutions to (P) and (D2)

respectively and the conditions of Theorem 23 are satisfiedFurther assume that the following conditions are satisfied

(i) Resum119896

119894=1119894[119891(120577 120578

119894) + 119911119867119860119908] is strict (119865 120572

1 1205881 120579)-

pseudoconvex with respect to 119877+on 119876 and ℎ(120577) is

(119865 1205722 1205882 120579)-quasiconvex on 119876 with respect to the

polyhedral cone 119878 sub 119862119901(ii) 12058811205721( ) + 120588

21205722( ) ge 0

Then 120577 = 120585 that is 120585 is an optimal solution to (D2)

Proof On the contrary we assume that ( ) = 120577 = 120585 = ( )On applyingTheorem 23 we know that

supisin119884

Re [119891 (120577 120578) + (119867

119860)12

]

= supisin119884

Re [119891 (120585 120578) + (119867

119860)12

]

(99)

From the feasibility of 120577 isin 119876 to (P) 120583 isin 119878lowast inequality (81)yields

Re ⟨ℎ (120577) 120583⟩ le 0 le Re ⟨ℎ (120585) 120583⟩ (100)

Since ℎ(120577) is (119865 1205722 1205882 120579)-quasiconvex on 119876 with respect to

the polyhedral cone 119878 in 119862119901 the above inequality yields

119865 [ 1205722( ) (120583

119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

le minus12058821205792

( )

(101)

Journal of Optimization 11

which by sublinearity of 119865 implies

119865 [ (120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

le minus1205882

1205722( )

1205792

( )

(102)

By (80) and the sublinearity of 119865 we have

119865[

[

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]]

]

+ 119865 [ (120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

ge 119865[

[

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]

+ 120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585)]

]

= 0

(103)

The above inequality together with (102) and 12058811205721( ) +

12058821205722( ) ge 0 gives

119865[

[

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]]

]

ge minus119865 [ (120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

ge1205882

1205722( )

1205792

( )

ge minus1205881

1205721( )

1205792

( )

(104)

That is

119865[

[

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]]

]

ge minus1205881

1205721( )

1205792 ( )

(105)

which by sublinearity of 119865 implies

119865[

[

1205721( )

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]]

]

ge minus12058811205792 ( )

(106)

Since Resum119896

119894=1119894[119891(120577 120578

119894) + 119911119867119860119908] is strict (119865 120572

1 1205881 120579)-

pseudoconvex with respect to 119877+on 119876 the above inequality

implies that

Re[

[

119896

sum119894=1

119894[119891 (120577 120578

119894) + 119867

119860119908]]

]

gt Re[

[

119896

sum119894=1

119894[119891 (120585 120578

119894) + 119867

119860119908]]

]

(107)

From (82) (83) and the generalized Schwarz inequality weget

Re (119867119860119908) le (119867

119860)12

Re (119867119860119908)12

= (119867

119860)12

(108)

which on substituting in (107) we obtain

Re[

[

119896

sum119894=1

119894[119891 (120577 120578

119894) + (

119867

119860)12

] ]

]

gt Re[

[

119896

sum119894=1

119894[119891 (120585 120578

119894) + (

119867

119860)12

]]

]

(109)

Consequently there exist certain 1198940which satisfy

Re [119891 (120577 1205781198940) + (119867119860)

12

]

gt Re [119891 (120585 1205781198940) + (119867119860)

12

]

(110)

Hence

supisin119884

Re [119891 (120577 120578) + (119867

119860)12

]

ge Re [119891 (120577 1205781198940) + (

119867

119860)12

]

gt Re [119891 (120585 1205781198940) + (

119867

119860)12

]

= supisin119884

Re [119891 (120585 120578) + (119867

119860)12

]

(111)

which contradicts (99) hence the theorem

6 Conclusion

In this paper we introduced generalized (119865 120572 120588 120579)-convexfunctions and established sufficient optimality conditions fora class of nondifferentiable minimax programming problemsin complex space These optimality conditions are thenused to construct two types of dual model and finally wederived weak strong and strict converse duality theorems

12 Journal of Optimization

to show that there is no duality gap between the two dualproblems with respect to the primal problem under somegeneralized convexities of complex functions in the complexprogramming problem As a future task the authors wouldlike to extend these results to second and higher order casesand establish the relations between primal and its second andhigher order dual problems

Acknowledgments

The authors are thankful to the anonymous referees for theirvaluable comments which have improved the presentationof the paper The research of the first author is financiallysupported by the University Grant Commission New DelhiIndia through Grant No F no 41-8012012(SR)

References

[1] N Levinson ldquoLinear programming in complex spacerdquo Journalof Mathematical Analysis and Applications vol 14 no 1 pp 44ndash62 1966

[2] J C Chen and H C Lai ldquoOptimality conditions for minimaxprogramming of analytic functionsrdquoTaiwanese Journal ofMath-ematics vol 8 no 4 pp 673ndash686 2004

[3] N Datta and D Bhatia ldquoDuality for a class of nondifferentiablemathematical programming problems in complex spacerdquo Jour-nal of Mathematical Analysis and Applications vol 101 no 1 pp1ndash11 1984

[4] O P Jain and P C Saxena ldquoA duality theorem for a specialclass of programming problems in complex spacerdquo Journal ofOptimization Theory and Applications vol 16 no 3-4 pp 207ndash220 1975

[5] I M Stancu-Minasian Fractional Programming Theory Meth-ods and Applications vol 409 Kluwer Academic DordrechtGermany 1997

[6] I Ahmad and Z Husain ldquoOptimality conditions and dualityin nondifferentiable minimax fractional programming withgeneralized convexityrdquo Journal of Optimization Theory andApplications vol 129 no 2 pp 255ndash275 2006

[7] A Jayswal I Ahmad and D Kumar ldquoNondifferentiable mini-max fractional programming problem with nonsmooth gener-alized convex functionsrdquoCommunications onAppliedNonlinearAnalysis vol 18 no 4 pp 57ndash75 2011

[8] A Jayswal and D Kumar ldquoOn nondifferentiable minimaxfractional programming involving generalized (C 120572 120588 d)-convexityrdquo Communications on Applied Nonlinear Analysis vol18 no 1 pp 61ndash77 2011

[9] H-C Lai J-C Lee and S-C Ho ldquoParametric duality on mini-max programming involving generalized convexity in complexspacerdquo Journal of Mathematical Analysis and Applications vol323 no 2 pp 1104ndash1115 2006

[10] K Swarup and I C Sharma ldquoProgramming with linear frac-tional functionals in complex spacerdquo Cahiers Center drsquoEtudesRecherche Operationelle vol 12 pp 103ndash109 1970

[11] A Charnes and W W Cooper ldquoProgramming with linearfractional functionalsrdquo Naval Research Logistics Quarterly vol9 no 3-4 pp 181ndash186 1962

[12] H-C Lai and T-Y Huang ldquoOptimality conditions for non-differentiable minimax fractional programming with complexvariablesrdquo Journal of Mathematical Analysis and Applicationsvol 359 no 1 pp 229ndash239 2009

[13] H-C Lai and J-C Liu ldquoDuality for nondifferentiable minimaxprogramming in complex spacesrdquo Nonlinear Analysis TheoryMethods and Applications vol 71 no 12 pp e224ndashe233 2009

[14] I Ahmad Z Husain and S Sharma ldquoSecond-order dualityin nondifferentiable minmax programming involving type-Ifunctionsrdquo Journal of Computational and Applied Mathematicsvol 215 no 1 pp 91ndash102 2008

[15] I Ahmad Z Husain and S Sharma ldquoHigher-order duality innondifferentiableminimax programmingwith generalized typeI functionsrdquo Journal of Optimization Theory and Applicationsvol 141 no 1 pp 1ndash12 2009

[16] BMond andBD Craven ldquoA class of nondifferentiable complexprogramming problemsrdquo Journal of Mathematics and Statisticsvol 6 pp 581ndash591 1975

[17] S K Mishra and N G Rueda ldquoSymmetric duality for math-ematical programming in complex spaces with F-convexityrdquoJournal of Mathematical Analysis and Applications vol 284 no1 pp 250ndash265 2003

[18] H C Lai and J C Liu ldquoMinimax programming in complexspaces-necessary and sufficient optimality conditionsrdquo in Pro-ceedings of the International Symposium on Nonlinear Analysisand Convex Analysis 2007 Research Institute forMathematicalSciences Kyoto University Japan Rims Kokuroku 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Optimization 9

From (51) (52) and the generalized Schwarz inequality wehave

Re (119867119860119908) le (119867

119860)12

Re (119867119860119908) = (119867

119860)12

(76)

which on substituting in (75) and by using (49) we obtain

Re[

[

119896

sum119894=1

119894[119891 (120577 120578

119894) + (

119867

119860)12

]]

]

gt Re[

[

119896

sum119894=1

119894[119891 (120585 120578

119894) + (

119867

119860)12

]]

]

ge

119896

sum119894=1

119894

(77)

Consequently there exist certain 1198940which satisfy

Re [119891 (120577 1205781198940) + (

119867

119860)12

] gt (78)

Hence

supisin119884

Re [119891 (120577 120578) + (119867

119860)12

]

ge Re [119891 (120577 1205781198940) + (

119867

119860)12

] gt

(79)

which contradicts (67) hence the theorem

5 Parameter Free Duality

Making use of the optimality conditions we show thatthe following formation is a dual (D2) to the complexprogramming problem (P)

max(119896120582120578)isin119870(120585)

sup(120585120583119908)isin119883(119896120582120578)

Re [119891 (120585 120578) + (119906119867

119860119906)12

] (D2)

where119883(119896 120578) denotes the set of all (120585 120583 119908) isin 1198622119899times119862119901times119862119899

to satisfy the following conditions

119896

sum119894=1

120582119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]

+ 120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585) = 0

(80)

Re ⟨ℎ (120585) 120583⟩ ge 0 (81)

119908119867

119860119908 le 1 (82)

(119906119867

119860119906)12

= Re (119906119867119860119908) (83)

If for a triplet (119896 120578) isin 119870(120585) the set 119883(119896 120578) = 0then we define the supremum over 119883(119896 120578) to be minusinfin fornonexception in the formulation of (D2)

Nowwe establish appropriate duality theorems and provethat optimal values of (P) and (D2) are equal under theassumption of generalized convexity in order to show that theproblems (P) and (D2) have no duality gap

Theorem21 (weak duality) Let 120577 = (119911 119911) and (119896 120578 120585 120583 119908)

be feasible solutions to (P) and (D2) respectively Furtherif Re[sum119896

119894=1120582119894[119891(120577 120578

119894) + 119911119867119860119908]] is (119865 120572

1 1205881 120579)-pseudoconvex

with respect to 119877+on 119876 ℎ(120577) is (119865 120572

2 1205882 120579)-quasiconvex on 119876

with respect to the polyhedral cone 119878 sub 119862119901 and 12058811205721(z 119906) +

12058821205722(119911 119906) ge 0 then

sup120578isin119884

Re [119891 (120577 120578) + (119911119867

119860119911)12

]

ge sup120578isin119884

Re [119891 (120585 120578) + (119906119867

119860119906)12

]

(84)

Proof On the contrary we suppose that

sup120578isin119884

Re [119891 (120577 120578) + (119911119867

119860119911)12

]

lt sup120578isin119884

Re [119891 (120585 120578) + (119906119867

119860119906)12

]

(85)

Since 120578119894isin 119884(120585) sub 119884 119894 = 1 2 119896 we have

sup120578isin119884

Re [119891 (120585 120578) + (119906119867

119860119906)12

]

= Re [119891 (120585 120578119894) + (119906

119867

119860119906)12

] 119894 = 1 2 119896

(86)

Re [119891 (120577 120578119894) + (119911

119867

119860119911)12

]

le sup120578isin119884

Re [119891 (120577 120578) + (119911119867

119860119911)12

] 119894 = 1 2 119896

(87)

Then the above three inequalities give

Re [119891 (120577 120578119894) + (119911

119867

119860119911)12

]

lt Re [119891 (120585 120578119894) + (119906

119867

119860119906)12

] 119894 = 1 2 119896

(88)

From (82) (83) (88) and the generalized Schwarz inequalitywe have

Re [119891 (120577 120578119894) + (119911

119867

119860119908)]

lt Re [119891 (120585 120578119894) + (119906

119867

119860119908)] 119894 = 1 2 119896

(89)

As 120582119894gt 0 119894 = 1 2 119896 and sum

119896

119894=1120582119894= 1 we have

Re[119896

sum119894=1

120582119894[119891 (120577 120578

119894) + (119911

119867

119860119908)]]

minus Re[119896

sum119894=1

120582119894[119891 (120585 120578

119894) + (119906

119867

119860119908)]] lt 0

(90)

10 Journal of Optimization

Since Re[sum119896119894=1

120582119894[119891(120577 120578

119894) + 119911119867119860119908]] is (119865 120572

1 1205881 120579)-

pseudoconvex with respect to 119877+on 119876 the above inequality

implies that

119865[119911 119906 1205721(119911 119906)

times

119896

sum119894=1

120582119894nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]

lt minus12058811205792

(119911 119906)

(91)

which by sublinearity of 119865 becomes

119865[119911 119906

119896

sum119894=1

120582119894nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]

lt minus1205881

1205721(119911 119906)

1205792

(119911 119906)

(92)

By the feasibility of 120577 = (119911 119911) to (P) 0 = 120583 isin 119878lowast and theinequality (81) we obtain

Re ⟨ℎ (120577) 120583⟩ le 0 le Re ⟨ℎ (120585) 120583⟩ (93)

The above inequality together with the (119865 1205722 1205882 120579)-

quasiconvexity of ℎ(120577) on 119876 with respect to the polyhedralcone 119878 sub 119862

119901 implies

119865 [119911 119906 1205722(119911 119906) (120583

119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

le minus12058821205792

(119911 119906)

(94)

which by sublinearity of 119865 becomes

119865 [119911 119906 120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585)]

le minus1205882

1205722(119911 119906)

1205792

(119911 119906) (95)

On adding (92) and (95) and using the sublinearity of 119865 weget

119865[119911 119906

119896

sum119894=1

120582119894nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908

+120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585) ]

lt minus(1205881

1205721(119911 119906)

+1205882

1205722(119911 119906)

) 1205792

(119911 119906)

(96)

From the assumption 12058811205721(119911 119906) +120588

21205722(119911 119906) ge 0 the above

inequality yields

119865[119911 119906

119896

sum119894=1

120582119894nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908

+ 120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585) ] lt 0

(97)

which contradicts (80) hence the theorem

Theorem 22 (weak duality) Let 120577 = (119911 119911) and(119896 120578 120585 120583 119908) be feasible solutions to (P) and (D2)respectively Further if Re[sum119896

119894=1120582119894[119891(120577 120578

119894) + 119911

119867

119860119908]] is(119865 1205721 1205881 120579)-quasiconvex with respect to 119877

+on 119876 ℎ(120577) is

(119865 1205722 1205882 120579)-pseudoconvex on 119876 with respect to the polyhedral

cone 119878 sub 119862119901 and 12058811205721(119911 119906) + 120588

21205722(119911 119906) ge 0 then

sup120578isin119884

Re [119891 (120577 120578) + (119911119867

119860119911)12

]

ge sup120578isin119884

Re [119891 (120585 120578) + (119906119867

119860119906)12

]

(98)

Proof Theproof follows the same lines as inTheorem 21

Theorem 23 (strong duality) Let 1205770= (1199110 1199110) be an optimal

solution to the problem (P) at which a constraint qualificationis a satisfied Then there exist (119896 120578) isin 119870(120577

0) and (120577

0 120583 119908) isin

119883(119896 120578) such that (119896 120578 1205770 120583 119908) is a feasible solution to the

dual problem (D2) Further if the hypotheses of Theorem 21or Theorem 22 are satisfied then (119896 120578 120577

0 120583 119908) is optimal

to (D2) and the two problems (P) and (D2) have the sameoptimal values

Proof Theproof follows along the lines ofTheorem8 (Lai andLiu [13])

Theorem 24 (strict converse duality) Let 120577 and(

120582 120578 120585 120583 119908) be optimal solutions to (P) and (D2)

respectively and the conditions of Theorem 23 are satisfiedFurther assume that the following conditions are satisfied

(i) Resum119896

119894=1119894[119891(120577 120578

119894) + 119911119867119860119908] is strict (119865 120572

1 1205881 120579)-

pseudoconvex with respect to 119877+on 119876 and ℎ(120577) is

(119865 1205722 1205882 120579)-quasiconvex on 119876 with respect to the

polyhedral cone 119878 sub 119862119901(ii) 12058811205721( ) + 120588

21205722( ) ge 0

Then 120577 = 120585 that is 120585 is an optimal solution to (D2)

Proof On the contrary we assume that ( ) = 120577 = 120585 = ( )On applyingTheorem 23 we know that

supisin119884

Re [119891 (120577 120578) + (119867

119860)12

]

= supisin119884

Re [119891 (120585 120578) + (119867

119860)12

]

(99)

From the feasibility of 120577 isin 119876 to (P) 120583 isin 119878lowast inequality (81)yields

Re ⟨ℎ (120577) 120583⟩ le 0 le Re ⟨ℎ (120585) 120583⟩ (100)

Since ℎ(120577) is (119865 1205722 1205882 120579)-quasiconvex on 119876 with respect to

the polyhedral cone 119878 in 119862119901 the above inequality yields

119865 [ 1205722( ) (120583

119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

le minus12058821205792

( )

(101)

Journal of Optimization 11

which by sublinearity of 119865 implies

119865 [ (120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

le minus1205882

1205722( )

1205792

( )

(102)

By (80) and the sublinearity of 119865 we have

119865[

[

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]]

]

+ 119865 [ (120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

ge 119865[

[

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]

+ 120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585)]

]

= 0

(103)

The above inequality together with (102) and 12058811205721( ) +

12058821205722( ) ge 0 gives

119865[

[

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]]

]

ge minus119865 [ (120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

ge1205882

1205722( )

1205792

( )

ge minus1205881

1205721( )

1205792

( )

(104)

That is

119865[

[

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]]

]

ge minus1205881

1205721( )

1205792 ( )

(105)

which by sublinearity of 119865 implies

119865[

[

1205721( )

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]]

]

ge minus12058811205792 ( )

(106)

Since Resum119896

119894=1119894[119891(120577 120578

119894) + 119911119867119860119908] is strict (119865 120572

1 1205881 120579)-

pseudoconvex with respect to 119877+on 119876 the above inequality

implies that

Re[

[

119896

sum119894=1

119894[119891 (120577 120578

119894) + 119867

119860119908]]

]

gt Re[

[

119896

sum119894=1

119894[119891 (120585 120578

119894) + 119867

119860119908]]

]

(107)

From (82) (83) and the generalized Schwarz inequality weget

Re (119867119860119908) le (119867

119860)12

Re (119867119860119908)12

= (119867

119860)12

(108)

which on substituting in (107) we obtain

Re[

[

119896

sum119894=1

119894[119891 (120577 120578

119894) + (

119867

119860)12

] ]

]

gt Re[

[

119896

sum119894=1

119894[119891 (120585 120578

119894) + (

119867

119860)12

]]

]

(109)

Consequently there exist certain 1198940which satisfy

Re [119891 (120577 1205781198940) + (119867119860)

12

]

gt Re [119891 (120585 1205781198940) + (119867119860)

12

]

(110)

Hence

supisin119884

Re [119891 (120577 120578) + (119867

119860)12

]

ge Re [119891 (120577 1205781198940) + (

119867

119860)12

]

gt Re [119891 (120585 1205781198940) + (

119867

119860)12

]

= supisin119884

Re [119891 (120585 120578) + (119867

119860)12

]

(111)

which contradicts (99) hence the theorem

6 Conclusion

In this paper we introduced generalized (119865 120572 120588 120579)-convexfunctions and established sufficient optimality conditions fora class of nondifferentiable minimax programming problemsin complex space These optimality conditions are thenused to construct two types of dual model and finally wederived weak strong and strict converse duality theorems

12 Journal of Optimization

to show that there is no duality gap between the two dualproblems with respect to the primal problem under somegeneralized convexities of complex functions in the complexprogramming problem As a future task the authors wouldlike to extend these results to second and higher order casesand establish the relations between primal and its second andhigher order dual problems

Acknowledgments

The authors are thankful to the anonymous referees for theirvaluable comments which have improved the presentationof the paper The research of the first author is financiallysupported by the University Grant Commission New DelhiIndia through Grant No F no 41-8012012(SR)

References

[1] N Levinson ldquoLinear programming in complex spacerdquo Journalof Mathematical Analysis and Applications vol 14 no 1 pp 44ndash62 1966

[2] J C Chen and H C Lai ldquoOptimality conditions for minimaxprogramming of analytic functionsrdquoTaiwanese Journal ofMath-ematics vol 8 no 4 pp 673ndash686 2004

[3] N Datta and D Bhatia ldquoDuality for a class of nondifferentiablemathematical programming problems in complex spacerdquo Jour-nal of Mathematical Analysis and Applications vol 101 no 1 pp1ndash11 1984

[4] O P Jain and P C Saxena ldquoA duality theorem for a specialclass of programming problems in complex spacerdquo Journal ofOptimization Theory and Applications vol 16 no 3-4 pp 207ndash220 1975

[5] I M Stancu-Minasian Fractional Programming Theory Meth-ods and Applications vol 409 Kluwer Academic DordrechtGermany 1997

[6] I Ahmad and Z Husain ldquoOptimality conditions and dualityin nondifferentiable minimax fractional programming withgeneralized convexityrdquo Journal of Optimization Theory andApplications vol 129 no 2 pp 255ndash275 2006

[7] A Jayswal I Ahmad and D Kumar ldquoNondifferentiable mini-max fractional programming problem with nonsmooth gener-alized convex functionsrdquoCommunications onAppliedNonlinearAnalysis vol 18 no 4 pp 57ndash75 2011

[8] A Jayswal and D Kumar ldquoOn nondifferentiable minimaxfractional programming involving generalized (C 120572 120588 d)-convexityrdquo Communications on Applied Nonlinear Analysis vol18 no 1 pp 61ndash77 2011

[9] H-C Lai J-C Lee and S-C Ho ldquoParametric duality on mini-max programming involving generalized convexity in complexspacerdquo Journal of Mathematical Analysis and Applications vol323 no 2 pp 1104ndash1115 2006

[10] K Swarup and I C Sharma ldquoProgramming with linear frac-tional functionals in complex spacerdquo Cahiers Center drsquoEtudesRecherche Operationelle vol 12 pp 103ndash109 1970

[11] A Charnes and W W Cooper ldquoProgramming with linearfractional functionalsrdquo Naval Research Logistics Quarterly vol9 no 3-4 pp 181ndash186 1962

[12] H-C Lai and T-Y Huang ldquoOptimality conditions for non-differentiable minimax fractional programming with complexvariablesrdquo Journal of Mathematical Analysis and Applicationsvol 359 no 1 pp 229ndash239 2009

[13] H-C Lai and J-C Liu ldquoDuality for nondifferentiable minimaxprogramming in complex spacesrdquo Nonlinear Analysis TheoryMethods and Applications vol 71 no 12 pp e224ndashe233 2009

[14] I Ahmad Z Husain and S Sharma ldquoSecond-order dualityin nondifferentiable minmax programming involving type-Ifunctionsrdquo Journal of Computational and Applied Mathematicsvol 215 no 1 pp 91ndash102 2008

[15] I Ahmad Z Husain and S Sharma ldquoHigher-order duality innondifferentiableminimax programmingwith generalized typeI functionsrdquo Journal of Optimization Theory and Applicationsvol 141 no 1 pp 1ndash12 2009

[16] BMond andBD Craven ldquoA class of nondifferentiable complexprogramming problemsrdquo Journal of Mathematics and Statisticsvol 6 pp 581ndash591 1975

[17] S K Mishra and N G Rueda ldquoSymmetric duality for math-ematical programming in complex spaces with F-convexityrdquoJournal of Mathematical Analysis and Applications vol 284 no1 pp 250ndash265 2003

[18] H C Lai and J C Liu ldquoMinimax programming in complexspaces-necessary and sufficient optimality conditionsrdquo in Pro-ceedings of the International Symposium on Nonlinear Analysisand Convex Analysis 2007 Research Institute forMathematicalSciences Kyoto University Japan Rims Kokuroku 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Journal of Optimization

Since Re[sum119896119894=1

120582119894[119891(120577 120578

119894) + 119911119867119860119908]] is (119865 120572

1 1205881 120579)-

pseudoconvex with respect to 119877+on 119876 the above inequality

implies that

119865[119911 119906 1205721(119911 119906)

times

119896

sum119894=1

120582119894nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]

lt minus12058811205792

(119911 119906)

(91)

which by sublinearity of 119865 becomes

119865[119911 119906

119896

sum119894=1

120582119894nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]

lt minus1205881

1205721(119911 119906)

1205792

(119911 119906)

(92)

By the feasibility of 120577 = (119911 119911) to (P) 0 = 120583 isin 119878lowast and theinequality (81) we obtain

Re ⟨ℎ (120577) 120583⟩ le 0 le Re ⟨ℎ (120585) 120583⟩ (93)

The above inequality together with the (119865 1205722 1205882 120579)-

quasiconvexity of ℎ(120577) on 119876 with respect to the polyhedralcone 119878 sub 119862

119901 implies

119865 [119911 119906 1205722(119911 119906) (120583

119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

le minus12058821205792

(119911 119906)

(94)

which by sublinearity of 119865 becomes

119865 [119911 119906 120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585)]

le minus1205882

1205722(119911 119906)

1205792

(119911 119906) (95)

On adding (92) and (95) and using the sublinearity of 119865 weget

119865[119911 119906

119896

sum119894=1

120582119894nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908

+120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585) ]

lt minus(1205881

1205721(119911 119906)

+1205882

1205722(119911 119906)

) 1205792

(119911 119906)

(96)

From the assumption 12058811205721(119911 119906) +120588

21205722(119911 119906) ge 0 the above

inequality yields

119865[119911 119906

119896

sum119894=1

120582119894nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908

+ 120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585) ] lt 0

(97)

which contradicts (80) hence the theorem

Theorem 22 (weak duality) Let 120577 = (119911 119911) and(119896 120578 120585 120583 119908) be feasible solutions to (P) and (D2)respectively Further if Re[sum119896

119894=1120582119894[119891(120577 120578

119894) + 119911

119867

119860119908]] is(119865 1205721 1205881 120579)-quasiconvex with respect to 119877

+on 119876 ℎ(120577) is

(119865 1205722 1205882 120579)-pseudoconvex on 119876 with respect to the polyhedral

cone 119878 sub 119862119901 and 12058811205721(119911 119906) + 120588

21205722(119911 119906) ge 0 then

sup120578isin119884

Re [119891 (120577 120578) + (119911119867

119860119911)12

]

ge sup120578isin119884

Re [119891 (120585 120578) + (119906119867

119860119906)12

]

(98)

Proof Theproof follows the same lines as inTheorem 21

Theorem 23 (strong duality) Let 1205770= (1199110 1199110) be an optimal

solution to the problem (P) at which a constraint qualificationis a satisfied Then there exist (119896 120578) isin 119870(120577

0) and (120577

0 120583 119908) isin

119883(119896 120578) such that (119896 120578 1205770 120583 119908) is a feasible solution to the

dual problem (D2) Further if the hypotheses of Theorem 21or Theorem 22 are satisfied then (119896 120578 120577

0 120583 119908) is optimal

to (D2) and the two problems (P) and (D2) have the sameoptimal values

Proof Theproof follows along the lines ofTheorem8 (Lai andLiu [13])

Theorem 24 (strict converse duality) Let 120577 and(

120582 120578 120585 120583 119908) be optimal solutions to (P) and (D2)

respectively and the conditions of Theorem 23 are satisfiedFurther assume that the following conditions are satisfied

(i) Resum119896

119894=1119894[119891(120577 120578

119894) + 119911119867119860119908] is strict (119865 120572

1 1205881 120579)-

pseudoconvex with respect to 119877+on 119876 and ℎ(120577) is

(119865 1205722 1205882 120579)-quasiconvex on 119876 with respect to the

polyhedral cone 119878 sub 119862119901(ii) 12058811205721( ) + 120588

21205722( ) ge 0

Then 120577 = 120585 that is 120585 is an optimal solution to (D2)

Proof On the contrary we assume that ( ) = 120577 = 120585 = ( )On applyingTheorem 23 we know that

supisin119884

Re [119891 (120577 120578) + (119867

119860)12

]

= supisin119884

Re [119891 (120585 120578) + (119867

119860)12

]

(99)

From the feasibility of 120577 isin 119876 to (P) 120583 isin 119878lowast inequality (81)yields

Re ⟨ℎ (120577) 120583⟩ le 0 le Re ⟨ℎ (120585) 120583⟩ (100)

Since ℎ(120577) is (119865 1205722 1205882 120579)-quasiconvex on 119876 with respect to

the polyhedral cone 119878 in 119862119901 the above inequality yields

119865 [ 1205722( ) (120583

119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

le minus12058821205792

( )

(101)

Journal of Optimization 11

which by sublinearity of 119865 implies

119865 [ (120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

le minus1205882

1205722( )

1205792

( )

(102)

By (80) and the sublinearity of 119865 we have

119865[

[

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]]

]

+ 119865 [ (120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

ge 119865[

[

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]

+ 120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585)]

]

= 0

(103)

The above inequality together with (102) and 12058811205721( ) +

12058821205722( ) ge 0 gives

119865[

[

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]]

]

ge minus119865 [ (120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

ge1205882

1205722( )

1205792

( )

ge minus1205881

1205721( )

1205792

( )

(104)

That is

119865[

[

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]]

]

ge minus1205881

1205721( )

1205792 ( )

(105)

which by sublinearity of 119865 implies

119865[

[

1205721( )

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]]

]

ge minus12058811205792 ( )

(106)

Since Resum119896

119894=1119894[119891(120577 120578

119894) + 119911119867119860119908] is strict (119865 120572

1 1205881 120579)-

pseudoconvex with respect to 119877+on 119876 the above inequality

implies that

Re[

[

119896

sum119894=1

119894[119891 (120577 120578

119894) + 119867

119860119908]]

]

gt Re[

[

119896

sum119894=1

119894[119891 (120585 120578

119894) + 119867

119860119908]]

]

(107)

From (82) (83) and the generalized Schwarz inequality weget

Re (119867119860119908) le (119867

119860)12

Re (119867119860119908)12

= (119867

119860)12

(108)

which on substituting in (107) we obtain

Re[

[

119896

sum119894=1

119894[119891 (120577 120578

119894) + (

119867

119860)12

] ]

]

gt Re[

[

119896

sum119894=1

119894[119891 (120585 120578

119894) + (

119867

119860)12

]]

]

(109)

Consequently there exist certain 1198940which satisfy

Re [119891 (120577 1205781198940) + (119867119860)

12

]

gt Re [119891 (120585 1205781198940) + (119867119860)

12

]

(110)

Hence

supisin119884

Re [119891 (120577 120578) + (119867

119860)12

]

ge Re [119891 (120577 1205781198940) + (

119867

119860)12

]

gt Re [119891 (120585 1205781198940) + (

119867

119860)12

]

= supisin119884

Re [119891 (120585 120578) + (119867

119860)12

]

(111)

which contradicts (99) hence the theorem

6 Conclusion

In this paper we introduced generalized (119865 120572 120588 120579)-convexfunctions and established sufficient optimality conditions fora class of nondifferentiable minimax programming problemsin complex space These optimality conditions are thenused to construct two types of dual model and finally wederived weak strong and strict converse duality theorems

12 Journal of Optimization

to show that there is no duality gap between the two dualproblems with respect to the primal problem under somegeneralized convexities of complex functions in the complexprogramming problem As a future task the authors wouldlike to extend these results to second and higher order casesand establish the relations between primal and its second andhigher order dual problems

Acknowledgments

The authors are thankful to the anonymous referees for theirvaluable comments which have improved the presentationof the paper The research of the first author is financiallysupported by the University Grant Commission New DelhiIndia through Grant No F no 41-8012012(SR)

References

[1] N Levinson ldquoLinear programming in complex spacerdquo Journalof Mathematical Analysis and Applications vol 14 no 1 pp 44ndash62 1966

[2] J C Chen and H C Lai ldquoOptimality conditions for minimaxprogramming of analytic functionsrdquoTaiwanese Journal ofMath-ematics vol 8 no 4 pp 673ndash686 2004

[3] N Datta and D Bhatia ldquoDuality for a class of nondifferentiablemathematical programming problems in complex spacerdquo Jour-nal of Mathematical Analysis and Applications vol 101 no 1 pp1ndash11 1984

[4] O P Jain and P C Saxena ldquoA duality theorem for a specialclass of programming problems in complex spacerdquo Journal ofOptimization Theory and Applications vol 16 no 3-4 pp 207ndash220 1975

[5] I M Stancu-Minasian Fractional Programming Theory Meth-ods and Applications vol 409 Kluwer Academic DordrechtGermany 1997

[6] I Ahmad and Z Husain ldquoOptimality conditions and dualityin nondifferentiable minimax fractional programming withgeneralized convexityrdquo Journal of Optimization Theory andApplications vol 129 no 2 pp 255ndash275 2006

[7] A Jayswal I Ahmad and D Kumar ldquoNondifferentiable mini-max fractional programming problem with nonsmooth gener-alized convex functionsrdquoCommunications onAppliedNonlinearAnalysis vol 18 no 4 pp 57ndash75 2011

[8] A Jayswal and D Kumar ldquoOn nondifferentiable minimaxfractional programming involving generalized (C 120572 120588 d)-convexityrdquo Communications on Applied Nonlinear Analysis vol18 no 1 pp 61ndash77 2011

[9] H-C Lai J-C Lee and S-C Ho ldquoParametric duality on mini-max programming involving generalized convexity in complexspacerdquo Journal of Mathematical Analysis and Applications vol323 no 2 pp 1104ndash1115 2006

[10] K Swarup and I C Sharma ldquoProgramming with linear frac-tional functionals in complex spacerdquo Cahiers Center drsquoEtudesRecherche Operationelle vol 12 pp 103ndash109 1970

[11] A Charnes and W W Cooper ldquoProgramming with linearfractional functionalsrdquo Naval Research Logistics Quarterly vol9 no 3-4 pp 181ndash186 1962

[12] H-C Lai and T-Y Huang ldquoOptimality conditions for non-differentiable minimax fractional programming with complexvariablesrdquo Journal of Mathematical Analysis and Applicationsvol 359 no 1 pp 229ndash239 2009

[13] H-C Lai and J-C Liu ldquoDuality for nondifferentiable minimaxprogramming in complex spacesrdquo Nonlinear Analysis TheoryMethods and Applications vol 71 no 12 pp e224ndashe233 2009

[14] I Ahmad Z Husain and S Sharma ldquoSecond-order dualityin nondifferentiable minmax programming involving type-Ifunctionsrdquo Journal of Computational and Applied Mathematicsvol 215 no 1 pp 91ndash102 2008

[15] I Ahmad Z Husain and S Sharma ldquoHigher-order duality innondifferentiableminimax programmingwith generalized typeI functionsrdquo Journal of Optimization Theory and Applicationsvol 141 no 1 pp 1ndash12 2009

[16] BMond andBD Craven ldquoA class of nondifferentiable complexprogramming problemsrdquo Journal of Mathematics and Statisticsvol 6 pp 581ndash591 1975

[17] S K Mishra and N G Rueda ldquoSymmetric duality for math-ematical programming in complex spaces with F-convexityrdquoJournal of Mathematical Analysis and Applications vol 284 no1 pp 250ndash265 2003

[18] H C Lai and J C Liu ldquoMinimax programming in complexspaces-necessary and sufficient optimality conditionsrdquo in Pro-ceedings of the International Symposium on Nonlinear Analysisand Convex Analysis 2007 Research Institute forMathematicalSciences Kyoto University Japan Rims Kokuroku 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Optimization 11

which by sublinearity of 119865 implies

119865 [ (120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

le minus1205882

1205722( )

1205792

( )

(102)

By (80) and the sublinearity of 119865 we have

119865[

[

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]]

]

+ 119865 [ (120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

ge 119865[

[

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]

+ 120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585)]

]

= 0

(103)

The above inequality together with (102) and 12058811205721( ) +

12058821205722( ) ge 0 gives

119865[

[

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]]

]

ge minus119865 [ (120583119879

nabla119911ℎ (120585) + 120583

119867

nabla119911ℎ (120585))]

ge1205882

1205722( )

1205792

( )

ge minus1205881

1205721( )

1205792

( )

(104)

That is

119865[

[

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]]

]

ge minus1205881

1205721( )

1205792 ( )

(105)

which by sublinearity of 119865 implies

119865[

[

1205721( )

119896

sum119894=1

119894[nabla119911119891 (120585 120578

119894) + nabla119911119891 (120585 120578

119894) + 119860119908]]

]

ge minus12058811205792 ( )

(106)

Since Resum119896

119894=1119894[119891(120577 120578

119894) + 119911119867119860119908] is strict (119865 120572

1 1205881 120579)-

pseudoconvex with respect to 119877+on 119876 the above inequality

implies that

Re[

[

119896

sum119894=1

119894[119891 (120577 120578

119894) + 119867

119860119908]]

]

gt Re[

[

119896

sum119894=1

119894[119891 (120585 120578

119894) + 119867

119860119908]]

]

(107)

From (82) (83) and the generalized Schwarz inequality weget

Re (119867119860119908) le (119867

119860)12

Re (119867119860119908)12

= (119867

119860)12

(108)

which on substituting in (107) we obtain

Re[

[

119896

sum119894=1

119894[119891 (120577 120578

119894) + (

119867

119860)12

] ]

]

gt Re[

[

119896

sum119894=1

119894[119891 (120585 120578

119894) + (

119867

119860)12

]]

]

(109)

Consequently there exist certain 1198940which satisfy

Re [119891 (120577 1205781198940) + (119867119860)

12

]

gt Re [119891 (120585 1205781198940) + (119867119860)

12

]

(110)

Hence

supisin119884

Re [119891 (120577 120578) + (119867

119860)12

]

ge Re [119891 (120577 1205781198940) + (

119867

119860)12

]

gt Re [119891 (120585 1205781198940) + (

119867

119860)12

]

= supisin119884

Re [119891 (120585 120578) + (119867

119860)12

]

(111)

which contradicts (99) hence the theorem

6 Conclusion

In this paper we introduced generalized (119865 120572 120588 120579)-convexfunctions and established sufficient optimality conditions fora class of nondifferentiable minimax programming problemsin complex space These optimality conditions are thenused to construct two types of dual model and finally wederived weak strong and strict converse duality theorems

12 Journal of Optimization

to show that there is no duality gap between the two dualproblems with respect to the primal problem under somegeneralized convexities of complex functions in the complexprogramming problem As a future task the authors wouldlike to extend these results to second and higher order casesand establish the relations between primal and its second andhigher order dual problems

Acknowledgments

The authors are thankful to the anonymous referees for theirvaluable comments which have improved the presentationof the paper The research of the first author is financiallysupported by the University Grant Commission New DelhiIndia through Grant No F no 41-8012012(SR)

References

[1] N Levinson ldquoLinear programming in complex spacerdquo Journalof Mathematical Analysis and Applications vol 14 no 1 pp 44ndash62 1966

[2] J C Chen and H C Lai ldquoOptimality conditions for minimaxprogramming of analytic functionsrdquoTaiwanese Journal ofMath-ematics vol 8 no 4 pp 673ndash686 2004

[3] N Datta and D Bhatia ldquoDuality for a class of nondifferentiablemathematical programming problems in complex spacerdquo Jour-nal of Mathematical Analysis and Applications vol 101 no 1 pp1ndash11 1984

[4] O P Jain and P C Saxena ldquoA duality theorem for a specialclass of programming problems in complex spacerdquo Journal ofOptimization Theory and Applications vol 16 no 3-4 pp 207ndash220 1975

[5] I M Stancu-Minasian Fractional Programming Theory Meth-ods and Applications vol 409 Kluwer Academic DordrechtGermany 1997

[6] I Ahmad and Z Husain ldquoOptimality conditions and dualityin nondifferentiable minimax fractional programming withgeneralized convexityrdquo Journal of Optimization Theory andApplications vol 129 no 2 pp 255ndash275 2006

[7] A Jayswal I Ahmad and D Kumar ldquoNondifferentiable mini-max fractional programming problem with nonsmooth gener-alized convex functionsrdquoCommunications onAppliedNonlinearAnalysis vol 18 no 4 pp 57ndash75 2011

[8] A Jayswal and D Kumar ldquoOn nondifferentiable minimaxfractional programming involving generalized (C 120572 120588 d)-convexityrdquo Communications on Applied Nonlinear Analysis vol18 no 1 pp 61ndash77 2011

[9] H-C Lai J-C Lee and S-C Ho ldquoParametric duality on mini-max programming involving generalized convexity in complexspacerdquo Journal of Mathematical Analysis and Applications vol323 no 2 pp 1104ndash1115 2006

[10] K Swarup and I C Sharma ldquoProgramming with linear frac-tional functionals in complex spacerdquo Cahiers Center drsquoEtudesRecherche Operationelle vol 12 pp 103ndash109 1970

[11] A Charnes and W W Cooper ldquoProgramming with linearfractional functionalsrdquo Naval Research Logistics Quarterly vol9 no 3-4 pp 181ndash186 1962

[12] H-C Lai and T-Y Huang ldquoOptimality conditions for non-differentiable minimax fractional programming with complexvariablesrdquo Journal of Mathematical Analysis and Applicationsvol 359 no 1 pp 229ndash239 2009

[13] H-C Lai and J-C Liu ldquoDuality for nondifferentiable minimaxprogramming in complex spacesrdquo Nonlinear Analysis TheoryMethods and Applications vol 71 no 12 pp e224ndashe233 2009

[14] I Ahmad Z Husain and S Sharma ldquoSecond-order dualityin nondifferentiable minmax programming involving type-Ifunctionsrdquo Journal of Computational and Applied Mathematicsvol 215 no 1 pp 91ndash102 2008

[15] I Ahmad Z Husain and S Sharma ldquoHigher-order duality innondifferentiableminimax programmingwith generalized typeI functionsrdquo Journal of Optimization Theory and Applicationsvol 141 no 1 pp 1ndash12 2009

[16] BMond andBD Craven ldquoA class of nondifferentiable complexprogramming problemsrdquo Journal of Mathematics and Statisticsvol 6 pp 581ndash591 1975

[17] S K Mishra and N G Rueda ldquoSymmetric duality for math-ematical programming in complex spaces with F-convexityrdquoJournal of Mathematical Analysis and Applications vol 284 no1 pp 250ndash265 2003

[18] H C Lai and J C Liu ldquoMinimax programming in complexspaces-necessary and sufficient optimality conditionsrdquo in Pro-ceedings of the International Symposium on Nonlinear Analysisand Convex Analysis 2007 Research Institute forMathematicalSciences Kyoto University Japan Rims Kokuroku 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Journal of Optimization

to show that there is no duality gap between the two dualproblems with respect to the primal problem under somegeneralized convexities of complex functions in the complexprogramming problem As a future task the authors wouldlike to extend these results to second and higher order casesand establish the relations between primal and its second andhigher order dual problems

Acknowledgments

The authors are thankful to the anonymous referees for theirvaluable comments which have improved the presentationof the paper The research of the first author is financiallysupported by the University Grant Commission New DelhiIndia through Grant No F no 41-8012012(SR)

References

[1] N Levinson ldquoLinear programming in complex spacerdquo Journalof Mathematical Analysis and Applications vol 14 no 1 pp 44ndash62 1966

[2] J C Chen and H C Lai ldquoOptimality conditions for minimaxprogramming of analytic functionsrdquoTaiwanese Journal ofMath-ematics vol 8 no 4 pp 673ndash686 2004

[3] N Datta and D Bhatia ldquoDuality for a class of nondifferentiablemathematical programming problems in complex spacerdquo Jour-nal of Mathematical Analysis and Applications vol 101 no 1 pp1ndash11 1984

[4] O P Jain and P C Saxena ldquoA duality theorem for a specialclass of programming problems in complex spacerdquo Journal ofOptimization Theory and Applications vol 16 no 3-4 pp 207ndash220 1975

[5] I M Stancu-Minasian Fractional Programming Theory Meth-ods and Applications vol 409 Kluwer Academic DordrechtGermany 1997

[6] I Ahmad and Z Husain ldquoOptimality conditions and dualityin nondifferentiable minimax fractional programming withgeneralized convexityrdquo Journal of Optimization Theory andApplications vol 129 no 2 pp 255ndash275 2006

[7] A Jayswal I Ahmad and D Kumar ldquoNondifferentiable mini-max fractional programming problem with nonsmooth gener-alized convex functionsrdquoCommunications onAppliedNonlinearAnalysis vol 18 no 4 pp 57ndash75 2011

[8] A Jayswal and D Kumar ldquoOn nondifferentiable minimaxfractional programming involving generalized (C 120572 120588 d)-convexityrdquo Communications on Applied Nonlinear Analysis vol18 no 1 pp 61ndash77 2011

[9] H-C Lai J-C Lee and S-C Ho ldquoParametric duality on mini-max programming involving generalized convexity in complexspacerdquo Journal of Mathematical Analysis and Applications vol323 no 2 pp 1104ndash1115 2006

[10] K Swarup and I C Sharma ldquoProgramming with linear frac-tional functionals in complex spacerdquo Cahiers Center drsquoEtudesRecherche Operationelle vol 12 pp 103ndash109 1970

[11] A Charnes and W W Cooper ldquoProgramming with linearfractional functionalsrdquo Naval Research Logistics Quarterly vol9 no 3-4 pp 181ndash186 1962

[12] H-C Lai and T-Y Huang ldquoOptimality conditions for non-differentiable minimax fractional programming with complexvariablesrdquo Journal of Mathematical Analysis and Applicationsvol 359 no 1 pp 229ndash239 2009

[13] H-C Lai and J-C Liu ldquoDuality for nondifferentiable minimaxprogramming in complex spacesrdquo Nonlinear Analysis TheoryMethods and Applications vol 71 no 12 pp e224ndashe233 2009

[14] I Ahmad Z Husain and S Sharma ldquoSecond-order dualityin nondifferentiable minmax programming involving type-Ifunctionsrdquo Journal of Computational and Applied Mathematicsvol 215 no 1 pp 91ndash102 2008

[15] I Ahmad Z Husain and S Sharma ldquoHigher-order duality innondifferentiableminimax programmingwith generalized typeI functionsrdquo Journal of Optimization Theory and Applicationsvol 141 no 1 pp 1ndash12 2009

[16] BMond andBD Craven ldquoA class of nondifferentiable complexprogramming problemsrdquo Journal of Mathematics and Statisticsvol 6 pp 581ndash591 1975

[17] S K Mishra and N G Rueda ldquoSymmetric duality for math-ematical programming in complex spaces with F-convexityrdquoJournal of Mathematical Analysis and Applications vol 284 no1 pp 250ndash265 2003

[18] H C Lai and J C Liu ldquoMinimax programming in complexspaces-necessary and sufficient optimality conditionsrdquo in Pro-ceedings of the International Symposium on Nonlinear Analysisand Convex Analysis 2007 Research Institute forMathematicalSciences Kyoto University Japan Rims Kokuroku 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of