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

I, ' IlK f ,,r h i

S

(II (d

9

'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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