ft !h e t. it ; d Corr t I r
(IJ-32)
IF- 7'- 7
i-fhNNf tA ITINjHP A[\kP/IfTPY
I I rin r iT i
I [f RA I VI ,01II ION Of L INI AR PROGRAM'>
by
(i. I . M(inlltjrarian
t,ppIl~r i d t.hen ait. " i 5Di v inn
Jull y 1979
Nutllt 1
1 9, vp I t a, I,,9,,l 999f ,*,"Int 9. w~IIn e gi r.; athr 1 cutr 9' i r a. all 9
gIn , itll & fit ,I ihie. ,,a, va99r 1 1 t, n t+. nen ruh ,,nts Ir "rte rnl~ ,cak
IIA",I ' ~r e tp,1mrh9 I, i u '
p9tr~r lts I rrri ll ,. h.ifvi99a" "1*
1a9191991 y ' , ,? r eI o t - -
Research supported by Nat. ionalAO? and MC 7901,)0 9Ind in part
Science Foundat inn Grants MC /4-?H5814by the . . Department of Fnrgy.
Current addrtsr : Cromputer sciences Depart.m nt. andIndustrial Fmqnineering, University of Wisconsin, 1Street, Madison, Wisconsin 53706.
I)epartment. of?10 West flayton
*
1*
TABLE E OF CONTENTS
Abstract . . . . . . .
1. Introduction . . .
. Iterative Lolution
3. Iterativr solution n
4. More General Linea
5. Numericd PesulLt,-
Ac know] edrjermen -t .
ef r ncE . . . . . .
. . . . . . . . . . . . . . . . . . .
of the Quadratic Programming Problem
of the Linear Programming P oblem . .
r Pr gram, . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . .
iii
Pale
iv
1
4
11
14
19
20
I If RAT IV I ,Ol UI T ION OF I !Nf AP PROGRAM-
fl. I. ManI -r 1 irin
l , , t, r ir.
f ir .1 I i v
1 e I At t. r
tli th 1 VO
ir pr rgJrd rl t o d rTI h J1 I t it rr( (Ir .iM, it i , )l 1(-
in i t, r11i I v Ir 1i 11h <pa( ' P y I t.ir at iV( I'r hl irqurs
oVer'-r(' 1 ix a t. ion (OOP ) rnr-tlod', . Th I rov i d a
inal i n ar Lr J( rnm.
iv
y -r i
to "o Ivi'
yJ I a"
" l t io l
l I ntrolucti)n
We shall be conCe-rned here with iterative wet hcds for so v ling the
linear program
Mini rize p Tx subject to Ax b
where p and b are given vectors in Rn and r1 res pectively, and
A is a (Ji ven m-n real m1atrix with no rows th t re ide iti cally
zero. The most populn' methods for solving this problem are direct
pivotal methods such as the simplex method and its variants (Refs 1-2).
However, more recently there have been nu;:ber of iterative procedures
proposed (Refs. 3-8). Some of these (Refs. 3-5) consist of an
iterative method for finding a feasible point of the Karu.>h-Kuhn-Tucker
inequalities that constitute the optical ity conditions of the linear
prorjran (1). Others (Pefs. 6 and 81 consist of minirizing a non smooth
reformulated problem: or (Pef, 7) finding stationary points of an
augmented Lagrangian. Our approach here is different, We consider
the follo1 inj qu d ra tic progran;ing perturbation of (1)
Minimize x- : 4 pTx subject to Ax b
It has been shown (Ref. 9) that (2) has a unique solution x, for
all in (0,7] for some E > 0, which is independent of r and
which also solves the linear program (1). By working in the dual
variable space of (2), we can utilize thk iterative techniq'ies
developed in (Rief. 10) 'or solving the symmetric linear complermen-
tarity problem to solve (2). It turns out that a Aufficient condition
(1)
(2)
for the iterative method to lead to a solution is that the constraints
of the linear program (1) be stable (Ref. 11), whi, h in this case
means that they satisfy the Slater constraint qualification (Ref., 12),
It is interesting to note that in order to obtain convergence of the
present, i ter ati ve method , use is made of various recent re sil ts con-
cerning linear program ., namely (a) nonlinear perturbation of linear
programs (Ref. 9) which orig-inated with th- uniqueness characterization
of linear programming solutions (,ef 13), (b) stability of sy-temis
of linear inequal ities (Ref 11) and (c) general sufficient conditions
for the convergence of iterative techniques for the solution of the
syimumetric 1 in r2ar co!iyple>,ntar ity problem (Ref. 10).
The out.] Me of the ,per is as fol los In Section 2 we
describe how our proposed iterative procedure is applied to a (eneral
qua-ratic program with a positive definite Hnssian, This procedure
may be used also in finding the projection of a point on a polytope.
In Section 3 we adapt the procedure of Section 2 specifically for
sol vinq the 1 inear program (1) by solving the perturbed quadratic
program (2). In Section i we give an algorithm for the solution of
more general 1 inear pro; a' .. In Section 5 we present some numerical
results whi ch show that when the perturbation and relaxation pararate rs
are properly chosen our proposed iterative method is competitive with
the revised simplex r(ethod (Refs. 1-2) and may even be more robust in
that it can solve problem, for which a revised simplex code fails.
We briefly describe now the notation used in this paper. All
matrices and vectors are real. For an mxn matrix A, row i is
denoted by A. and the element in row i and column j by A. .
For x in the real n-dimens ional Euclidean space k1, clement
is denoted by x . The superscript T denotes the transpose. All
vectors are column vectors unless transposed. Superscripts such as
i iK , u refer to specific reitrices and vectors and usually denote
iteration numbers. If u is in km, u+ denotes the vector in Rm
with elements
(u+)i = Mdx {o,uit, i 1,....,m
The vector e wil
denotes the m-m
vector x in Rn
1 denote a vector of ones in Rm or
identity matrix. The [uclidean norm
will be denoted by Ix1.
ln, and I
T '(x x) of a
4
2. Iterati v Solution of the quadratic_ PrograTminrg Problem
We sha.ll consider in this section the quadratic program
Minimize z xTQx + p x subject to Ax b (3)
where p, b, Q and A are respectively a vector in Rn, a vector
in F*m, real syrimetric positive definite n n matrix and a real
m,.*n matri x. We sha ll devel op an iterative al gori thn for sol ving
the dual of (3) (Ref. 12) :
Maximize - xTQx + bTu subject to Qx - ATu 4 p = 0, u > 0 (4)x,u
which under the positive definite assumption on Q is, upon elimi-
nating x, equivalent to
Minimize 1 IT -IATu - (b+AQlp)Tu (5)u -0
The proposed iterative procedure will solve (5), and from the
solution u the solution x of (3) is then obtained from
x = Q~ 1 (A Tu--p) (6)
Because tne proposed procedure involves the inversion of the matrix
Q, it is not, in general, a suitable procedure for solving (3) with
a general Q. However, for certain applications such as those
requiring the solution of (2), Q is the diagonal matrix cI, which
is easily inverted. Another such problem is that of projecting a
point c in Rn on the polytope {xIAx>h} using the Euclidean
norm, in which case Q = I and p = -C.
We are now ready to apply the results of Ref.
iteratively once we realize that, since AQ~iAT
definite,
is positive semi-
(5) is equivalent to the following symetric linear
cor'plementarity problem :
v = AQ~IATu - (b+AQ 1 p) 0, u
We shall use the following special case of Algorithm 2.1
Remark 2 4 of Ref. 10.
Ha vi rig u
1 Let
computei
u be an arbitrary nonnegative vector in
i +1u as follows :
ui~1 = (ui -(AQ~IA ui-b-AQ~ paK (u - u ) ))+
where E is a positive diagonal matrix, K1 is either the strict
lower triangular part L or the strict upper triangular part
AQ~IAT G is the diagonal of AQ~ AT
U of
and
0< <2/ max G..E..G. .>0
JJ
Note that for computing purposes
i+l i-1 i+1u1 , l2 '' '' . um
iwhen K =L
(9)
i+lu is computed in the order
and in the reverse of that
order when Kr = U.
The following theorem is a direct consequence of Theorem 2.1 of
Ref. 10.
0, uTv = 0
Algoritir 2
(7)
and
(8)
10 to solve (5)
Theorer 2.1 Let r, be syuv;etri c aind po i t i w dlef iri to. Thrn , each
accuult.i on po int u of the serlenr n u : nienerited1 b Al ;ri thm .1
oi ve, ( r) nd the crrep'0 ,i(Irinrn x deterroinei by (C) i the uriiiuO
mlution of (3)
This, th.or(Im doe not ju rintr:e the exi',tener: of a Lciculul tiorn
of the ,E ruiL(2e U , whredi the fr)l 1 i n'; (an doe, under the
,i'ih t 1 / more demnild in condi t of a ( rl tin i t rjual if i cA Li0n!.
h-orrn 2 L' 1. be ?/;,' tri. c nrid Err it i ,; dr f ini
cor'tr ii. of (3) Inti fy the 1ia (ter ucn trin t ou"1
is Ax S for 00H x ifn P 1 . 1h t'He cqeLric
Al. ir, i t hi 1 i 0ourd d a hd (1 h i t ledA ot one (i c ( ui ul
[ch ccu :ul 1 ti 0n point u of {u 1olv (5), aid
x determined by (G) i the uni que 0( 'olutio n cf (3).
t e, and 1 t the
ificitiOn, that
Sui (eniratr(d by
AtijOn point.
t.hr correopondig
P ronf n.cauoe Ax b, there exi
{ x Ax b 4 e i nonempty. E t
prograul
Minimize 1 xTQx + p Tx
sts L Cl 0 :ouch that
be the -ol uti On of the
subject to Ax : b 4 he
A solut i ~ to this problem exists becLuse Q is positive definite
and together with d u in Rm satisfies the following Karush-Kuhn-Tucker
condition (Ref. 1?) :
Qx4 p - ATu 0, u> 0, A > b + e,U (Ax-b-6e)= 0
the set
quCidratic
(AQ ) )T -
By iUrLr L G rid condi Lion (10) of Ref. 10
by Th.r r 1 ?1
abuvc , E C.h d.( . i i i t1 f
x dterri rid by (0) 1 thue
ReuUyrl ?, 1
E.1 Ves ( ) dii) the (corr .potidiri.J
ni(,ue m luti in of (3)
Tb' ' ter constraint qual ificatior i equivalent to
ter2 , t, b iitj c onditi on (Pef
e/i)t- (/., )
1.) that ftr each d in R"m
in r 1
Ax b) + rd, 0
Henrve
(bL 4.-1p)
is
there
bounded at he at lt:ast one accumulation point.
ui i
ati sfyingJ
.
3. Iter dliv ,1 ut i (in of ti' I in ',r I'ro rr- in', Probil ':
We now turn our fttrnntion hdck to the li e r prurd (1) dnd
stuto d rr alt whi h i" f direc.t c.onmu:(enio: of Theore, 1 of ;ef 9.
In-or_(r' 3.1 I et. the I inedr pro'r n (1) have ( ,0lutin.
dxi t a r l f f', iti : n ,ri r iuIJ I t fr(r ojd i.
(, t he. un i y , lt ion x of (?) i 1idepiendint of
a01 ut.ion of the 1 inenr pro;rrai (I).
I rnjm t hr pr f (f Thior(mi I (;f 11of. 9 we r n oh 2in
a poSt(ri(r i ul r hound on of 'th 'buv: th 'r -, ,r
wh r' ( i. the pO i Li vr op t il I :(r'n(j0 mu] t ip ior ds
witL th O 1 A.. c mniL trI int, of thle p)rohlemin
I hn n t.
ir. th f r
arid
r,.ry ,
soc i (ted
intr rv 1
1
.t . . 1 TMlnlml/r A x
sub.ject to Ax b, p - -
(Ind ;hre is the iir m,; of prob1m (1) If r 0 , then can
be (ny nionnvgAtive number . There is a] another int. resti 1n(1
interprebt ijon of - (:eI. 14). Ii we take the dual (Pef. 1?) of
the convex quidri tic program (2) , we obt a in the probl er
Miaximin e 1b1 u - - II u-p I 2 subject to u 0
This i pr i sely the exterior pondl ty problem (soci ated with the
dual 1 inar irIrograilm oif (1):
Mil ximi e hTu
with j)entdl ty pidrv9t(er
subject to ATu u 0
-- Results of ordinary exterior penaltyr
me thoos (Rof. I ) require
snr 1r rF ',ul ts that tdke
(P I. '16) requi .: erel
reuIt'. of (Pef 9).
W'( c An (Y 1r tline 1
11ro)l UT (2) 1' ) 1 ,
thf lrineir <rwrn (1).
S 1, F .U~ 1 he
folly 1. .wT
that t - , and hence c 0. However,
at Vanlta";e of 1 inI arity of th(: pro[W e
thi t for some .0 Or eq(Ui va l(Itly
These sharp:r resul t- corrcpuond to the cited
heore2 3.1 with Al jori thi 2 1 to -o lvJ
('i,] dl thui' obtain a ol ution to
In p1 rt i Cur we m t ini Al (or i thml ? 1,
i i the ( dgo aljria of PAA., and obt( in the
.A. '.'ri h
U in R .
1 r n ( a
.iiuV ti ill)i t i U
and any nonnegati vr vector
) follow :
u 1 ( u. (/,A ui-/yp- 10K (u -u )))
wh''re U i) the di a(0111l of A%) that i ,
U.. - A. (A.) , j 1 1,. .,ii J J
l i ii' r the strict lower tr i angul(ir part L or the strict
Tupper tri (ngI ul ar ia rt U of AIA , and 0 < 2.
Combininq Theoren, 3.1, 2,1 and 2. 2 , we obtain the following two
cOnve r c i (.e t heore;ms for Al go r i thr 3.1
Theorer' 3.? There exi,,t,, a real pcv itive nuinber I such that for
each f in the interval (0,7], each accuriulation point u of the
(10)
-I -
sequence fu } generated by Aljor i thm 3.1 sol ves
r'in1h]( iTT Tinim e u AA u - (Fb+Ap) uu;0
and the correspondinrj x which is independent of F and deter-
mined by
x (ATu-p)
is the unitiue solution of (2) and is also a solution of the linear
pro; rarn (1).
jaqain note that Theorem 3.? does not ijurmrantu. the existence
of an dccu:ul tion point where the folloving theor droe, under
the additional assuniption that the constraints are stable.
Theorem 3.3 Let Ax b for sor( x in R.
positive number such that for, each i in
the sequere f u } generated by Al cjori thm 3.1
leat one accumulation point. Each accumuliti
solves (11) a id t h c.orrosponding x, which is
and dLtermined by (12), i t~h uniquT solutions
solution of the linear program (1).
The)-, exists a real
the interval (0,],
is bounded and has at
(m point u of {u }
independent t of
of (?) and is also a
(11)
(12)
11
4. More General Linear Prc irims
We outline in this part of the paper the corresponding results
for the case of nr'vre general constraints and owit the proofs which
are si;iil; r to tho,,e of kef. 10 and of action 3 of this paper. lb
particular we consider here instead of (1) the 1 inear program
subject to Ax b, Cx = d
where the addi tioal:1 equality constraint is
kn matrix C the vector d in k .
encouipa se s f fineCdr proqramc of d very qener
assume, for si ity, that no rows of A
zero and associate 0iLh (13) the foll(oing
positive L ;
Minimniz x x + p x subjc2
specified through
We note that this
al type We shall
or C are identi
quadratic program
t to Ax b, Cx = d
and the corres pondi nq dual probl ei
i k 1 (u)T ( (AT CT) (u) + ('\p T u
u-0
where the relation between x, u and v is given by
x = fi (ATuCTv-p)
The iterative procedure associated with (15) is as follows.
Minimize p x (13)
the
problem
again
ally
for some
(14)
(15)
(16)
(hoor' , i p5i Li ve nuJr , an arbitrary non-Algori thrv 4.1r
neriative vr tor
in 1k. Hvin ()
and an frbitrjry vector
coiIpute
-1b~ (Q)where D i the di(j nalI of (A) ),
upper tri(rnjul r part of the same matrii,
is the tri c ty I/Ower or
0 -, 2 , and
u>\V)-J
1 heorr 1.1 here exi'.t', a rca]l pocj i ive rur rer K <,ur.h that for
e in the interval (0, ],
}edch accu!mulation point
Jenera ted ( by Al (ori thm 4 1 solves
uvj of the
correspond drkermined by (16), which is independent of
the unique so luti On of (14) and in (ddit ion is i sol ution of the
piro r'a m ( l ' )
Theoremu 4. If in add ition to the assumption-; of Theorem 4.1
constraints of (13) are stable -- that is, there rxi(st ; an
such that Ax - b, C^ d, and the rows of C are linearly independent --
then for in (0,7] the sequence { 1ii}bounded and hence has at least one accumulation
of Algorithm 4.1
(~)accumulation point solveS (15), and the corresponding
is
Each such
x , which is
0v
K v)(AT CT) (Ar PuI (b>kJ + i-+1
4
each
sequence u (1 ) . and t.hr.
, is
the
nx in I?
u in Rrri
fs o ll (J".
i
U+
V
indeperidernt of and dctermincd by (16), is the unique solution
of (14) ard is dlso a solution of the lirwar program (13).
5. Nuirirjl P lt
Soixe to A rt. re ul t,. were ob tai nrd u, i n the i te rei t, i
3.1 Itrtinj -l-h u 0 to ,o le the 1 irn(ar projr '
Argonri : fm in b r; I t) ry I I)
5 PLC 11 jrd the rnvr,rjtional1
purpjore e rev ', d 1iplex code
pW"l wV ': generated LiS fol1
i trix uithi rrnde el''en t A
[-100 ,i . . Tin- vector) b rn
n
. 1'J 1 '
1)
-1n
t 2 ~Jjl
vc 50R Al qori th-)
(1) on thre
'033 (.ri:putfer running: under VM rele(ee
'rn i torini q' .t, r,. for ci. rur tive
wie, al'eo 'lje, d (K:f. 17). The teet
owe.Th nmtri/ A wo a fullj dense
unifermily (Ii,,trihuted in th, interval
dI p wJere c~hOser surh th t
ni f x A .. -
J- 1
A ..1J
nif A. -' 0
J: I
i j 1,....,nl
and
j i J3 ""where J
n{i J A..
j 1 0
Thew cn i c e for b ind p mde the point x 2e satisfy the
constraint qu(1 ification A(2e) - b and the point x e prima'
noptimal with ( mIinimuII value of Y A... A dual optimal
j-1 1<J J
variable is given by u. 1 for i ( J and u. - 0 for i/J Resul ts
for six cases are summarized in Table 1. Note that for cases 1 , 2, 3
and 5, because n -m, the linear program (1) does not have a unique
0}, j =1,
solution (Ref. 13). Cases 4 and 6 hava a unique solution if lnd
only if the mtri , i th ro.: t., i. J, has 1 i nenrl y independent
colurs' (". 1 ). Thus the acn <y of the -olutions described
in Table 1 i .aSured by (a) th( siber of figure in agreermert
between ml mlulted objiect ve fBcti-n aI + thonrft.ic-!1
rinimu ; / , and (b ) thK -- rm ' t th in f s ibili ty Of1 i.J~
the lcul Lcd prii ul solution x, that is, E ma .'in 1 / (K.-A..x .
We now 'Le ne fol 1rsl'inq obh'rvtins rce(jrdino I-l 1
i) Excep t for eases 4 anid C, the corputinq tines fir the
two methods are qui te sir ildr. For ease 6, the revised
simplex I;ethod failed arid for 1 1 f. i tritivP method
took 2.5 tie:S !hei 2 ihK revi sd simpilx method.
(ii) The revised s i' ple, ml utions , when obtained, are more
accurate than those of the iterative method.
(iii The iterdtive method is more robust in the sense that it
never fail to provide some answer when the constraints
are s t:tl e.
(.v) T*he vdluIiEs of the pErturbat ion pararF ter and the
reliix1 tion pairmc.;i ter were oht(a i ned Lafter some experimenting,
but ci r not neces-sarily optional. Tabl e 2 gives a typical range
of answers obtained by the iterative method for various values
of F and r which led to the values f 105 and w = 0.5
given in Table 1 for the case of m := 250 and n = 100.
A5LE 1
'u-erical =esults fcr " i p x s.t. nx > b 4.s er i a n ~ fix
iterativo "'ciK'c
Case n
ce
2 50200.;
I CD 93
* C 250
Secords of
ne
6
9-
Accuracy
1o. of Correc: -',r" o
activee Function feas ib1 t
1I l.
gL4
23 15 0.218x1
/00aiied- Proble
Declared L'nsounded
Seccn-s r AcCUra cy
.f Correct vo ow ;cures in 07- ri n-
ective -Func:ion easi Ub47 C
I -
105 0.5 130C 70 -
I'-
9.5 106 13 0.960
-5 5' 1 0-6
evised Si :lex ''ethod
(' rrce r j c ; 1
it
- t
10 '0.
104 0.?
4 ,
105 0.
10 010~ 0.8
ReultL f
ercit ionr
70
500
500
500
500
1114
TABLE
or Itera tive [t hod f
r r~r~i, rof Vi rtual* chi no Time
* C?
64
6?
64
64
64
6?
130
2
or 'tin F
fNo. ofObject
x s. t. Ax.- b, m= 250, n 100
Accuracy
f Fi ure; in -Norm of Primal1ive Function Infe&ibi lity
-15 0.860 1r)
4 0.196 101
4 0.436 10
5 0.725 -1
4 0.164
4 0.116
4 0.8 101
10 0.484 -10-
i -,
(v)
(vi)
~ he
rrne thod i
COrmp' t i ti
cho ern n ,
a pp ror h
rob~u, tnr
lire in
15, P.
rathrIr
the ranrje mn - l- 12nn.
in Table 1 are all
215) refer to our iterative
than an under-relaxation
Tie vr: l ue'. of in Table 1
von thowih thr rel /a' tirn f
le" , thai 1, we r,'till (Pef.
r ethod , an over-rel /1 i on
rile thod
(b V(.! ri .'r i 1 r ul t', in d
a vi Ob : 1r Knf , v)frIi
ve w i th the re v i Ied .inpl f x
.xp" ,I ).' l byty :akinrj a few
aril ., . 0.5 (hn pi i i th
iO f t rye t. The ridin a d
,, ',iInplicity and ohitility to
r ;t, that the proposed i terati ve
rie hod These pa ramete rs aiy he
short te t run. tarti nn wi th
i+1ioe val ue', for whi h + / - x1I
,n t a g e ' , ( f t h e e t h o d a r e i t
Shranldl e 1arj p'r)bl
'1. .art .3 * E1r g)(,]r ' i niq ard /
' 1 r t r a] 'rrrlr ,r'! in ,
1n , inrc ( *f.,i t '/
.,4,4.
4.,,
4 4 ~ (7 ' ,rv, -
1 ,I t i r)
Int.
.4] ?2 .
., 'h h 1 ." ( (OI rf J ( . vJ ;'r ( -t t inn f tr nti ir
'f fOnvix t v >f ,t it b . 1 dy t rr In I
i. &i 41 , r . M., fee P 1 , I i It (i l M t h dofi (' f i n i n toh ( Cr mon 1 Po r't., Ind [pp 1 rI tion it I t ',o lt in of Prrbluon in Convex I'r
I"'1j) jtj 1i t1na I t l rf ; h t K an' e K1t' teiim t i I I'K ', ic ran~1at. i
F, . (" t ,o0 lv jI n
f'r
irfI t i t
tif Iua I
v Mt he I, iv I 1ng ne ,i n 1ar Pr eeitrams ;, Mat h It
,t t F.onvt rclnet. ita Pr eoqr airn
nd Ire ya1'ov, V. ., (.(r1 1"rn Ing l4 I r rt r 1 r nilr': irI 11nd1 I t f ((nr(II < lni rptit
in It 'rat.ive M( th(Irn ft rti n, I r.Onomic 1 , Mat
. F nionvi r I n n( P't.,) 4 (if ubtt rae ri t. ([pt imin /t inll ''r thod ,t hm a i u n (I n i i4I 1 II .
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14. Golsteini, F. G., Private Copimnunication, Moscow, IISSR, June 1978.
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