16 Ramu Dubey and Vishnu Narayan Mishra

1. Introduction

The fractional optimization problem with multiple objective functions havebeen the subject of intense investigations in the past few years, which haveproduced a number of optimality and duality results for these problems.Higher-order duality in non-linear programming has been studied in lastfew years by many researchers. In various numerical algorithms, higher-order duality is considered over first-order as it gives more closer bounds.

Higher-order duality in nonlinear programs have been studied by someresearchers. Mangasarian [8] formulated a class of higher-order dual prob-lems for the nonlinear programming problem ”min {f(x) : g(x) ≥ 0}”by introducing twice differentiable function h : Rn × Rn → R and k :Rn×Rn → Rm. The concept of higher-order convexity presented by Zhang[12] and derived duality results in multiobjective programming problem.Later on, Yang et al. [11] considered a unified higher-order dual model fornondifferentiable multiobjective programs and proved duality results un-der generalized assumptions. Suneja et al. [10] introduced a higher order(F,α, σ)-type I functions and formulated higher order dual programs for anondifferentiable multiobjective fractional programming problem.

Motivated by several concepts of generalized convexity, Gulati and Agar-wal [4] gave the concept of second-order -V-type I functions for multiob-jective programming problem which were recently extended to nondifferen-tiable case by Jayswal et al. [6]. Recently, Jayswal et al. [7] considerdedhigher-Order duality for multiobjective programming problems and dis-cussed duality theorems under (F,α, ρ, d) − V− type-I functions. Severalre searchers [[1], [2], [3], [9], [13]-[17]] have done their work in the relatedareas.

In this paper, we have generalized the definitions of higher-order pseudoquasi/ strictly pseudo quasi/weak strictly pseudo quasi- (V, ρ, d)-type-Ifunctions for a nondifferentiable multiobjective higher-order fractional pro-gramming problem. We have formulated higher-order unified dual and es-tablished duality results under higher order pseudo quasi/ strictly pseudoquasi/weak strictly pseudo quasi- (V, ρ, d)-type-I assumptions.

Nondifferentiable higher-order duality theorems for new type of ... 17

2. Preliminaries and Definitions

Throughout this paper, we use the index sets K = {1, 2, ..., k} and M ={1, 2, ...,m}.

Definition 2.1. Let Q ⊆ Rn be a compact convex set. The supportfunction of Q is defined by

s (y|Q) = max{yT z : z ∈ Q}.

Consider the following nondifferentiable multiobjective fractional program-ming problem:

(MFP)MinimizeΨ(x) =

µφ1(x) + s(x|C1)ψ1(x)− s(x|E1)

,φ2(x) + s(x|C2)ψ2(x)− s(x|E2)

, ...,φk(x) + s(x|Ck)

ψk(x)− s(x|Ek)

¶Tsubject to x ∈ Y 0 = {x ∈ Y : πj(x) + s(x|Dj) ≤ 0, j ∈M},

where Y ⊆ Rn is an open set. The functions φ, ψ : Y → Rk, π :Y → Rm are differentiable on Y and Ci, Ei, Dj are compact convexsets in Rn for i ∈ K and j ∈ M . Let φi(x) + s(x|Ci) ≥ 0 and ψi(x) −s(x|Ei) > 0, i ∈ K. Let H = (H1, H2, ...,Hk) : X × Rn → Rk andK = (K1, K2, ...,Km) : X ×Rn → Rm be differentiable functions, d : X ×X → R, z = (z1, z2, ..., zk), v = (v1, v2, ..., vk) and w = (w1, w2, ..., wm),where zi ∈ Ci, vi ∈ Ei and wj ∈ Dj , for i ∈ K and j ∈M. Let ρ = (ρ1, ρ2)such that ρ1 = (ρ1, ρ2, ..., ρk) ∈ Rk, ρ2 = (ρk+1, ρk+2, ..., ρk+m) ∈ Rm.

Definition 2.2. A point u ∈ Y 0 is efficient solution of (MFP) if ∃ nox ∈ Y 0 such that Ψ(x) ≤ Ψ(u).

Definition 2.3. ∀ i ∈ K, j ∈ M,

Ãφi(.) + (.)

T ziψi(.)− (.)T vi

, πj(.) + (.)Twj

!is

higher order pseudo quasi (V, ρ, d)-type I at u of (MFP) if, ∀ x ∈ Y 0 andp ∈ Rn, such that

18 Ramu Dubey and Vishnu Narayan Mishra

φi(x) + xT ziψi(x)− xT vi

<φi(u) + uT ziψi(u)− uT vi

+Hi(u, p)− pT∇pHi(u, p)

⇒ ηT (x, u)

(∇Ãφi(u) + uT ziψi(u)− uT vi

!+∇pHi(u, p)

)+ ρ1i d

2(x, u) < 0

and

−πj(u)− uTwj ≤ Kj(u, p)− pT∇pKj(u, p)

⇒ ηT (x, u){∇πj(u) +∇pKj(u, p)}+ ρ2jd2(x, u) ≤ 0.

Definition 2.4. ∀ i ∈ K, j ∈ M,

Ãφi(.) + (.)

T ziψi(.)− (.)T vi

, πj(.) + (.)Twj

!is

higher order strictly pseudo quasi (V, ρ, d)-type I at u of (MFP) if, ∀ x ∈ Y 0

and p ∈ Rn, such that

φi(x) + xT ziψi(x)− xT vi

≤ φi(u) + uT ziψi(u)− uT vi

+Hi(u, p)− pT∇pHi(u, p)

⇒ ηT (x, u)

(∇Ãφi(u) + uT ziψi(u)− uT vi

!+∇pHi(u, p)

)+ ρ1i d

2(x, u) < 0

and

−πj(u)− uTwj ≤ Kj(u, p)− pT∇pKj(u, p)

⇒ ηT (x, u){∇πj(u) +∇pKj(u, p)}+ ρ2jd2(x, u) ≤ 0.

Definition 2.5. ∀ i ∈ K, j ∈ M,

Ãφi(.) + (.)

T ziψi(.)− (.)T vi

, πj(.) + (.)Twj

!is

higher order weak strictly pseudo quasi (V, ρ, d)-type I at u of (MFP) if,∀ x ∈ Y 0 and p ∈ Rn, such that

φi(x) + xT ziψi(x)− xT vi

≤ φi(u) + uT ziψi(u)− uT vi

+Hi(u, p)− pT∇pHi(u, p)

Nondifferentiable higher-order duality theorems for new type of ... 19

⇒ ηT (x, u)

(∇Ãφi(u) + uT ziψi(u)− uT vi

!+∇pHi(u, p)

)+ ρ1i d

2(x, u) < 0

and

−πj(u)− uTwj ≤ Kj(u, p)− pT∇pKj(u, p)

⇒ ηT (x, u){∇πj(u) +∇pKj(u, p)}+ ρ2jd2(x, u) ≤ 0.

Definition 2.6. ∀ i ∈ K, j ∈M,

Ãφi(.) + (.)

T ziψi(.)− (.)T vi

, πj(.)+(.)Twj

!is higher

order quasi strictly pseudo (V, ρ, d)-type I at u of (MFP) if, ∀ x ∈ Y 0 andp ∈ Rn, such that

φi(x) + xT ziψi(x)− xT vi

≤ φi(u) + uT ziψi(u)− uT vi

+Hi(u, p)− pT∇pHi(u, p)

⇒ ηT (x, u)

(∇Ãφi(u) + uT ziψi(u)− uT vi

!+∇pHi(u, p)

)+ ρ1i d

2(x, u) ≤ 0

and

−πj(u)− uTwj ≤ Kj(u, p)− pT∇pKj(u, p)

⇒ ηT (x, u){∇πj(u) +∇pKj(u, p)}+ ρ2jd2(x, u) < 0.

Theorem 2.1 (K-K-T-type necessary condition)[5]. Let u be efficientsolution of (MFP) at which the Kuhn-Tucker constraint qualification issatisfied on X. Then, ∃ 0 < λ ∈ Rk, 0 ≤ yj ∈ Rm, zi ∈ Rn, vi, wj ∈Rn, i ∈ K, j ∈M such that

kXi=1

λi∇Ãφi(u) + uT ziψi(u)− uT vi

!+

mXj=1

yj∇(πj(u) + uT wj) = 0,(2.1)

mXj=1

yj(πj(u) + uT wj) = 0,(2.2)

20 Ramu Dubey and Vishnu Narayan Mishra

uT zi = S(u|Ci), uT vi = S(u|Ei), u

T wj = S(u|Dj),(2.3)

zi ∈ Ci, vi ∈ Di, wj ∈ Ej , i ∈ K, j ∈M.(2.4)

In the following section, we consider the following mixed higher-order

dual for (MFP) and derive duality theorems. The notationφ(.) + (.)T z

ψ(.)− (.)T v +

µTJ0

³πjJ0 + (.)

TwJ0´e denotes the vector whose components are

φ1(.) + (.)T z1

ψ1(.)− (.)T v1+X

j∈J0µj³πj + (.)

Twj

´,

φ2(.) + (.)T z2

ψ2(.)− (.)T v2+Xj∈J0

µj³πj + (.)

Twj

´, ...,

φk(.) + (.)T zk

ψk(.)− (.)T vk+Xj∈J0

µj³πj + (.)

Twj

´and {π + (.)Tw}µJβ denotes r-dimensional vector whose components areX

j∈J1{πj + (.)Twj},

Xj∈J2

{πj + (.)Twj}, ...,Xj∈Jr

{πj + (.)Twj}.

3. Unified higher-order duality model:

In this section, we formulate the following unified higher- order dual for(MFP) and establish duality theorems:

(HMDP) : Maximize

Ãφ1(y) + yT z1ψ1(y)− yT v1

+H1(y, p)−pT∇pH1(y, p)+Xj∈J0

µj{πj(y)+

yTwj +Kj(y, p)− pT∇pKj(y, p)}, ...,φk(y) + yT zkψk(y)− yT vk

+Hk(y, p)−pT∇pHk(y, p)+Xj∈J0

µj{πj(y)+yTwj+Kj(y, p)−pT∇pKj(y, p)}!

subject to y ∈ Y ,

kXi=1

λi

(∇Ãφi(y) + yT ziψi(y)− yT vi

!+∇pHi(y, p)

)(3.1)

Nondifferentiable higher-order duality theorems for new type of ... 21

+mXj=1

µj{∇πj(y) + wj +∇pKj(y, p)} = 0,

Xj∈Jβ

µj{πj(y) + yTwj +Kj(y, p)− pT∇pKj(y, p)} ≥ 0, β = 1, ..., r,(3.2)

λi ≥ 0,kXi=1

λi = 1,(3.3)

µj ≥ 0, zi ∈ Ci, vi ∈ Ei, wj ∈ Dj for i ∈ K, j ∈M,(3.4)

where Jδ ⊆ N , δ = 0, 1, ..., r withSrδ=0 Jδ = N and Jδ1

TJδ2 if δ1 6= δ2.

It may be noted that J0 = N and Jβ = φ(1 ≤ β ≤ r), we obtain Wolfe typedual. If J0 = φ, J1 = N and Jβ = φ (2 ≤ β ≤ r), then (HMDP) reducesto Mond-Weir Type dual.

Let W 0 be feasible solution of (HMDP).

Theorem 3.1 (Weak Duality Theorem). Let x ∈ Y 0 and (y, λ, v, µ, z, w, p) ∈W 0. Let

(i)

Ãφi(.) + (.)

T ziψi(.)− (.)T vi

+ µTJ0

³πjJ0 + (.)

TwjJ0´e, {πj(.) + (.)Twj}µJβ

!be higher-

order weak strictly pseudo quasi (V, ρ, d)-type I at y,

(ii)kXi=1

λiρ1i +

rXj=1

µjρ2j ≥ 0.

Then, the following cannot hold

φi(x) + s(x|Ci)

ψi(x)− s(x|Ei)≤ φi(y) + yT zi

ψi(y)− yT vi+Hi(y, p)− pT∇pHi(y, p)

+Xj∈J0

µj{πj(y) + yTwj +Kj(y, p)− pT∇pKj(y, p)},∀ i ∈ K(3.5)

22 Ramu Dubey and Vishnu Narayan Mishra

and

φj(x) + s(x|Cj)

ψj(x)− s(x|Ej)<

φj(y) + yT zjψj(x)− yT vj

+Hj(y, p)− pT∇pHj(y, p)

+Xj∈J0

µj{πj(y) + yTwj +Kj(y, p)− pT∇pKj(y, p)}, forsomej ∈ K

. (3.6)

Proof: If possible, then suppose inequalities (3.5) and (3.6) hold. AsxT zi ≤ s(x|Ci), xT vi ≤ s(x|Ei), ∀i ∈ K and

Xj∈J0

µj(πj(x) + xTwj) ≤ 0,

using the inequalities and the dual constraint (3.2), hypothesis (i) gives

ηT (x, y)

(∇Ãφi(y) + yT ziψi(y)− yT vi

!+∇pHi(y, p) +

Xj∈J0

µj{∇πj(y) + wj +∇pKj(y, p)}e)

< −ρ1d2(x, y)

and

ηT (x, y)Xj∈Jβ

µj{∇πj(y) + wj +∇pKj(y, p)}+ ρ2βd2(x, y) ≤ 0, β = 1, ..., r.

Since λ ≥ 0, λT e = 1, it follows that

ηT (x, y)

ÃkXi=1

λi

(∇Ãφi(y) + yT ziψi(y)− yT vi

!+∇pHi(y, p)

)

+Xj∈J0

µj{∇πj(y) + wj +∇pKj(y, p)}!< −

kXi=1

λiρ1i d2(x, y)

and

ηT (x, y)

à Xj∈Jβ

µj{∇πj(y) + wj +∇pKj(y, p)}!≤ −ρ2βd2(x, y), β = 1, ..., r.

Nondifferentiable higher-order duality theorems for new type of ... 23

Above inequalities follows that

ηT (x, y)

ÃkXi=1

λi

(∇Ãφi(y) + yT ziψi(y)− yT vi

!+∇pHi(y, p)

)+

mXj=1

µj{∇πj(y)+wj

+∇pKj(y, p)}!= ηT (x, y)

ÃkXi=1

λi

(∇Ãφi(y) + yT ziψi(y)− yT vi

!+∇pHi(y, p)

)

+Xj∈J0

µj{∇πj(y) + wj +∇pKj(y, p)}+Xj∈J1

µj{∇πj(y) + wj +∇pKj(y, p)}

+...+Xj∈Jr

µj{∇πj(y) + wj +∇pKj(y, p)}!

ηT (x, y)

ÃkXi=1

λi

(∇Ãφi(y) + yT ziψi(y)− yT vi

!+∇pHi(y, p)

)+

mXj=1

µj{∇πj(y)+wj

+∇pKj(y, p)}!≤ ηT (x, y)

ÃkXi=1

λi

(∇Ãφi(y) + yT ziψi(y)− yT vi

!+∇pHi(y, p)

)

+Xj∈J0

µj{∇πj(y) + wj +∇pKj(y, p)}+Xj∈J1

µj{∇πj(y) + wj

+∇pKj(y, p)}+ ...+Xj∈Jr

µj{∇πj(y) + wj +∇pKj(y, p)}!

< −

⎛⎝ kXi=1

λiρ1i +

rXj=1

µjρ2j

⎞⎠ d2(x, y).

Further, using hypothesis (ii), we have

ηT (x, y)

ÃkXi=1

λi

(∇Ãφi(y) + yT ziψi(y)− yT vi

!+∇pHi(y, p)

!)

+mXj=1

µj{∇πj(y) + wj +∇pKj(y, p)}!< 0,

which contradicts (3.1). Hence, completes the proof.

24 Ramu Dubey and Vishnu Narayan Mishra

Theorem 3.2 (Weak Duality Theorem). Let x ∈ Y 0 and (y, λ, v, µ, z, w, p) ∈W 0. Let

(i)

Ãφi(.) + (.)

T ziψi(.) + (.)T vi

+ µTJ0

³πjJ0 + (.)

TwjJ0´e, {πj(.) + (.)Twj}µJβ

!be higher-

order pseudo quasi (V, ρ, d)-type I at y,

(ii)kXi=1

λiρ1i +

rXj=1

µjρ2j ≥ 0.

Then, the following cannot hold

φi(x) + s(x|Ci)

ψi(x)− s(x|Ei)≤ φi(y) + yT zi

ψi(y)− yT vi+Hi(y, p)− pT∇pHi(y, p)

+Xj∈J0

µj{πj(y) + yTwj +Kj(y, p)− pT∇pKj(y, p)}, ∀ i ∈ K(3.7)

and

φj(x) + s(x|Cj)

ψj(x)− s(x|Ej)<

φj(y) + yT zjψj(x)− yT vj

+Hj(y, p)− pT∇pHj(y, p)

+Xj∈J0

µj{πj(y) + yTwj +Kj(y, p)− pT∇pKj(y, p)},for some j ∈ K.

(3.8)

Proof The proof follows on the lines of Theorem 3.1.

Theorem 3.3 (Strong Duality Theorem). If u is an efficient solutionof (MFP) and let the Kuhn-Tucker constraint qualification be satisfied.Then, ∃ λ ∈ Rk, y ∈ Rm, zi ∈ Rn, vi ∈ Rn and wj ∈ Rn, i ∈ K, j ∈ M,such that (u, z, v, y, λ, w, p) ∈ W 0 and the (MFP)and (HMDP) have equalvalues. Also, if

H(u, 0) = 0, ∇pH(u, 0) = 0, K(u, 0) = 0, ∇pK(u, 0) = 0.

Nondifferentiable higher-order duality theorems for new type of ... 25

Furthermore, if the assumptions of Theorem 3.1 or 3.2 hold for Y 0 andW 0, then (u, z, v, y, λ, w, p = 0) is an efficient solution of (HMDP).

Proof. Since u is an efficient solution for (MFP) and the Slaters con-straint qualification is satisfied, from Theorem 2.1, there exist 0 < λ ∈Rk, 0 ≤ yj ∈ Rm, zi ∈ Rn, vi, wj ∈ Rn, i ∈ K, j ∈M such that

kXi=1

λi∇Ãφi(u) + uT ziψi(u)− uT vi

!+

mXj=1

yj∇(πj(u) + uT wj) = 0,(3.9)

mXj=1

yj(πj(u) + uT wj) = 0,(3.10)

uT zi = S(u|Ci), uT vi = S(u|Ei), u

T wj = S(u|Dj),(3.11)

zi ∈ Ci, vi ∈ Di, wj ∈ Ej , i ∈ K, j ∈M.(3.12)

Using the assumptionH(u, 0) = 0, ∇pH(u, 0) = 0, K(u, 0) = 0, ∇pK(u, 0) =0, we find that (u, z, v, y, λ, w, p = 0) ∈ W 0 and the two objective valuesare same. With the help of contradiction, we can prove efficiency results.Hence, the results. 2

Theorem 3.4 (Strict Converse Duality Theorem). Let u ∈ Y 0 and(y, λ, µ, v, z, w, p) ∈W 0 such that

(i)kXi=1

λi

(φi(u) + uT ziψi(u)− uT vi

)≤

kXi=1

λi

(φi(y) + yT ziψi(y)− yT vi

+∇ρHi(y, p)− pT∇pHi(y, p)

+Xj∈J0

µj{πj(y) + yT wj +Kj(y, p)− pT∇pKj(y, p)

),

26 Ramu Dubey and Vishnu Narayan Mishra

(ii) ρ1i +rX

j=1

ρ2j ≥ 0, ∀ i, j,

(iii)

ÃkXi=1

λi

(φi(.) + (.)

T ziψi(.)− (.)T vi

)+Xj∈J0

µj {πj + (.)T wj}, {πj(.) + (.)T wj}µJβ

!

is higher order strictly pseudoquasi (V, ρ, d)− typeIat y.Then, u = y.

Proof. Suppose u 6= y. The dual constraint (3.2) and the hypothesis (iii),for β = 1, ..., r yield

ηT (u, y)Xj∈Jβ

µj{∇πj(y) +∇ρKj(y, p) + wj} ≤ −ρ2βd2(u, y)(3.13)

By the dual constraint (3.1), we have

ηT (u, y)

ÃkXi=1

λi

(∇Ãφi(u) + uT ziψi(u)− uT vi

!+∇pHi(y, p)

)

+mXj=1

µj

(∇πj(y) +∇pKj(y, p) + wj

)!= 0,

above inequalities with (3.13) give

ηT (u, y)

ÃkXi=1

λi

(∇Ãφi(u) + uT ziψi(u)− uT vi

!+∇pHi(y, p)

)+Xj∈J0

µj

(∇πj(y)

+∇pKj(y, p) + wj

)!≥ −ηT (u, y)

à Xj∈J1

µj{∇πj(y) +∇pKj(y, p) + wj}!

−...− ηT (u, y)(Xj∈Jr

µj{∇πj(y) +∇pKj(y, p) + wj}!

or

Nondifferentiable higher-order duality theorems for new type of ... 27

ηT (u, y)

ÃkXi=1

λi

(∇Ãφi(u) + uT ziψi(u)− uT vi

!+∇pHi(y, p)

!)

+Xj∈J0

µj

(∇πj(y) +∇pKj(y, p) + wj

)!≥

rXj=1

ρ2jd2(u, y),

by hypothesis (ii),

ηT (u, y)

ÃkXi=1

λi

(∇Ãφi(u) + uT ziψi(u)− uT vi

!+∇pHi(y, p)

)

+Xj∈J0

µj

(∇πj(y) +∇pKj(y, p) + wj

)!≥ −ρ1i d2(u, y).

Therefore, hypothesis (iii) in view ofXj∈J0

µj{πj(u) + uT wj} ≤ 0 yields

kXi=1

λi

(φi(u) + uT ziψi(u)− uT vi

)>

kXi=1

λi

(φi(y) + yT ziψi(y)− yT vi

+∇pHi(y, p)− pT∇pHi(y, p)

)

+Xj∈J0

µj{πj(y) + yT wj +Kj(y, p)− pT∇pKj(y, p)},

which contradicts hypothesis (i). Hence, the result.

Theorem 3.5 (Strict Converse Duality Theorem). Let u ∈ Y 0 and(y, λ, µ, v, z, w, p) ∈W 0 such that

(i)kXi=1

λi

(φi(u) + uT ziψi(u)− uT vi

)≤

kXi=1

λi

(φi(y) + yT ziψi(y)− yT vi

+∇pHi(y, p)− pT∇pHi(y, p)

+Xj∈J0

µj{πj(y) + yT wj +Kj(y, p)− pT∇pKj(y, p)

),

(ii) ρ1i +rX

j=1

ρ2j ≥ 0, ∀ i, j,