The reformulation of nonlinear complementarity problems using the Fischer-Burmeister function

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P E R G A M O N Applied Mathematics Letters 12 (1999) 7-12

Applied

Mathematics

Letters

T h e R e f o r m u l a t i o n of N o n l i n e a r

C o m p l e m e n t a r i t y P r o b l e m s u s i n g t h e

F i s c h e r - B u r m e i s t e r F u n c t i o n

R. ANDREANI*

Department of Computer Science and Statistics, University of the State of S. Paulo (UNESP), C.P. 136, C E P 15054-000, S~Lo Jos6 do Rio Preto-SP, Brazil

andr eani0nimitz, dote. £b£1ce. unesp, br J. M A R T f N E Z f

Department of Applied Mathematics, I M E C C - U N I C A M P , University of Campinas, C P 6065, 13081-970 Campinas SP, Brazil

mart inez©ime, uni camp. br

(Received May 1998; revised and accepted November 1998)

A b s t r a c t - - A bounded-level-set result for a reformulation of the box-constrained variational inequality problem proposed recently by Facchinei, Fischer and Kanzow is proved. An application of this result to the (unbounded) nonlinear complementarity problem is suggested. (~ 1999 Elsevier Science Ltd. All rights reserved.

K e y w o r d s - - C o m p l e m e n t a r i t y , Unconstrained minimization, Reformulation.

1. M A I N R E S U L T S

Given F : R n --* R " , F E C I ( R n ) , t h e B o x - C o n s t r a i n e d Variational I n e q u a l i t y P r o b l e m ( B V I P ) consists on finding x E ~ such t h a t

( F ( x ) , z - x) >_ O, for all z E 12, (1) where ~ is t h e c o m p a c t b o x

= {x E R " I ~ < x < r } . (2) T h e N o n l i n e a r C o m p l e m e n t a r i t y P r o b l e m ( N C P ) is p r o b l e m (1) w h e n ~ = {x E R n I x > 0}.

Facchinei, Fischer a n d K a n z o w [1] p r o p o s e d a reform,llation of t h e B V I P as t h e following o p t i m i z a t i o n p r o b l e m :

n n

Minimize liE(x) - u + vii 2 + ~ ~(ui, x, - e~) 2 + ~ ~(vi, rs - x~) =, (3)

i = 1 i = l

*This author was supported by FAPESP (Grants 96/1552-0 and 96/12503- 0).

fThis author was supported by PRONEX, FAPESP (Grant 90-3724-6), CNPq and FAEP-UNICAMP. We are indebted to two anonymous referees for helpful comments.

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8 R. ANDREANI AND J. M. MARTiNEZ

w h e r e II" II is t h e E u c l i d i a n n o r m a n d ~ is t h e F i s c h e r - B u r m e i s t e r f u n c t i o n

~ ( a , b) = v ~ a 2 + b 2 - a - b, for all a, b E R.

T h e i m p o r t a n c e of t h e F i s c h e r - B u r m e i s t e r r e f o r m u l a t i o n of c o m p l e m e n t a r i t y p r o b l e m s lies on t h e fact t h a t t h e original p r o b l e m is r e d u c e d to a s i m p l e p r o b l e m for which m a n y effective t e c h n i q u e s exist. ( O f course, this c h a r a c t e r i s t i c is s h a r e d b y o t h e r r e f o r m u l a t i o n s t h a t have been p r o p o s e d in r e c e n t literature.) I f t h e n u m b e r of variables is large a n d t h e J a c o b i a n of F is not v e r y sparse, it is i n t e r e s t i n g to use m a t r i x - f r e e a l g o r i t h m s t o solve (3). I t is w o r t h m e n t i o n i n g t h a t i n t e r i o r - p o i n t t e c h n i q u e s t h a t do not rely on r e f o r m u l a t i o n s (see [2,3] a n d references therein) s e e m to be v e r y efficient for solving c o m p l e m e n t a r i t y p r o b l e m s , a t least w h e n h a n d l i n g (sparse) f a c t o r i z a t i o n s of m a t r i c e s is possible.

F r o m now on we call f ( x , u, v) the objective function of (3). It is e a s y to see t h a t f ( x * , u*, v*) = 0

if a n d o n l y if, x* is a solution of t h e B V I P . In [1] it was p r o v e d t h a t if (x*, u*, v*) is a s t a t i o n a r y p o i n t of f a n d F ' ( x * ) is a P 0 - m a t r i x it necessarily holds t h a t f ( x * , u*, v * ) = 0. T h e first result

of this n o t e will be t o p r o v e t h a t , below a critical value, t h e level sets of f are b o u n d e d . T H E O R E M 1. A s s u m e t h a t - c o < g~ < ri < co, [or a11 i = 1 , . . . , n a n d

( r i -

~)

< - - ' for all i = 1 , . . . , n.

T h e n the set S - { ( x , u , v ) 6 ]R 3n [ f ( x , u , v ) < a s} is bounded.

PROOF. L e t (x, u, v) b e such t h a t f ( x , u, v) < a s. S u p p o s e t h a t xi - gi < - c t . T h e n

T h i s implies t h a t f ( x , u, v) > a2. Therefore, if (x, u, v) 6 S we necessarily have t h a t

xi _> ~i - c~, for all i = 1 , . . . , n. (4) T h e s a m e r e a s o n i n g allows us t o p r o v e t h a t u i _ > - ~ , x i _ < r i - a , a n d v i _ > - ~ , (5) for all i = 1 , . . . , n . S u p p o s e n o w t h a t S is u n b o u n d e d . T h e r e f o r e S contains an u n b o u n d e d sequence (x k, u k, vk). k

B y (4),(5) this implies t h a t t h e r e exists i 6 { 1 , . . . , n } such t h a t u i -+ co or t h e r e exists i E { 1 , . . . , n} such t h a t v/k --+ oo. Consider t h e first case. We have t h a t

k k 2

a s _> ( [ F + v , )

S o ,

k k

Since I[F(xk)]~l is b o u n d e d , this implies t h a t v/k --, co. Analogously, if we a s s u m e v k --+ co we

necessarily o b t a i n t h a t u k --* co, too. So, we can a s s u m e t h a t t h e r e exists i 6 {1 . . . n} such k

t h a t b o t h u i ---* oo a n d v k --+ co. T a k i n g a n a p p r o p r i a t e subsequence, a s s u m e , w i t h o u t loss of k * T h e r e f o r e ,

generality, t h a t {x/k } is convergent, say, x i --+ x i .

lira + - -

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Reformulation of Nonlinear Complementarity

So,

;iiI

j/m--

(xf-t?i)

-u”

=&--x5.

Analogously, Now,

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(7) Therefore, by (6),(7), o2 2 (xf - eij2 + (~5 -Ti)2. ₍₈₎

But the minimum value of the right-hand side of (8) is (ri - !i)2/2. So, by the definition of o,

we arrived to a contradiction. _I

Counterexample

In this counterexample, we show that the previous result is sharp. That is to say, the level set defined by f (x, u, v) 5 p2 can be unbounded, where

p = min 1

(ri - ei), i = 1 -

4 ,,...ln. 1

Define F(x) = EX (E 1 0) and R = {z E W ] 0 5 x 5 1). So,

f(x, 21, v) = (EX - u +

v)2 +

(p(u,

x)2 +

cp(v, 1 - x)2. The sequence {y” E (l/2, k, k), k = 0, 1,2, . . . } is unbounded. However,

f(y") = (;-k+k)‘+2p

2

2. APPLICATION TO THE NCP

The Fischer-Burmeister function has been applied by many authors to the nonlinear complementarity problem NCP by means of the reformulation

Minimize 2 cp(xi, [F(x)]~)~.

i=l

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10 R . ANDREANI AND J . M . MARTiNEZ

See the references of [1]. If x* is a stationary point of (9) and F ( x * ) is a P0-matrix, it can be ensured t h a t x* is a solution of the NCP. If F is a uniform P-function it can also be proved t h a t the objective function of (9) has bounded level sets, but this property could not hold under weaker assumptions. In fact, consider the N C P defined by F ( x ) = 1 - e -x. This function is strictly monotone and 0 is a solution. However, the level sets o f the function (9) are not bounded. In fact, for all x > 0,

q o ( x , F ( x ) ) 2 = 2 + ( 1 _ e - ~ ) 2 - ( x + l - e -x < 1 .

So, it is natural to ask whether the bounded-level-set result proved in the previous section can help to establish bounded-level-set reformulations of the N C P using the Fischer-Burmeister function.

Let us define L = ( L , . . . , L) • R = and consider the box

aL = { x • R = l0 < x < L } .

T h e N C P is naturally connected with the BVIP defined by F and

~'~L.

In the Facchinei-Fischer- Kanzow reformulation, the corresponding objective function is

n ' n

= - ~o (u~, x i ) + ~ qo (vi, L - x~) 2 (10)

f(x,u,v) liE(x) u+vl[2+5-~.

2

i = 1 i = 1

Clearly, f ( 0 , 0, 0) = [[F(0)[[ 2. Therefore, if we take L > [[F(0)[[2/2, T h e o r e m 1 guarantees that

{x • R n I f ( x , u, v) <_ f(O, O, 0)} is bounded.

This implies t h a t standard unconstrained minimization algorithms, which usually generate sequences satisfying f ( x k+1, u k+l, v k+l) < f ( x k, u k, v k) for all k will generate bounded sequences, if (x °, u °, v °) = (0, 0, 0). As a consequence, algorithms of t h a t class will find stationary points, which, under the assumptions of [1], will be solutions of the BVIP defined by F and 12L. So, in order to solve the N C P we only need to guarantee t h a t solutions of this BVIP are solutions of the NCP. An answer to this question is given in the following theorem.

THEOREM 2. Assume that ~ there exists a solution of the N C P that belongs to

~L

and that, for all x, y • R n,

([F (x)]i - [ f (y)]i) (xi - y,) _< 0, for all i = 1 , . . . , n

implies that

([F (x)]i - [F (Y)]i) (xi - Yi) = 0, for all / = 1 , . . . , n.

Then, every stationary point o f f (defined by (10)) is a solution of the NCP.

PROOF. Let x* be a stationary point of f . T h e condition imposed to F in the hypothesis implies t h a t

max {(Fi (x) - Fi (y)) (xi - yi)} _ 0.

i:xi~yi

Therefore, by T h e o r e m 5.8 of [3], F is a is P0-function and F'(x) is a P0-matrix for all x • R n. In particular, F l ( x *) is a P0-matrix. So, by [1], x* is a solution of the B V I P defined by ~L- T h a t i s

[E (x*)]i >_ 0, if x[ = 0, (11)

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R e f o r m u l a t i o n of N o n l i n e a r C o m p l e m e n t a r i t y i 1

and

IF (x*)] i _< 0, if x ; = L. Assume t h a t £ is a solution of the NCP. This implies t h a t

IF (x)]i -> 0, if ~i = 0 and [F (x)]i = 0, By (11)-(14) we have t h a t (13) if £i > 0. (14) ( [ F - IF ( x ; - < 0, (15) for all i = 1 , . . . , n . Let us define Z = {i • { 1 , . . . , n } IF(z*)] i < 0, x* = L}. (16) Assume, by contradiction, t h a t Z ~ 0. Then there exists j • { 1 , . . . ,n} such t h a t [F(x*)]j < 0 and x j = L. Therefore,

0 > [F (x*)]j x; _> [ f (z*)]j x; - I f (x*)]j ~j - [F (~)]j x; + [F (~)]j ~j (17)

= ( I F - t F -

But (15) and (17) contradict the hypothesis of the theorem.

Therefore, Z -- 0. Since x* is a solution of the BVIP, this implies t h a t x* is a solution of the

NCP, as we wanted to prove. |

REMARKS. In the linear case ( F ( x ) = M x + q) the hypothesis of Theorem 2 means t h a t the matrix M is column-sufficient. If F is monotone ( ( F ( x ) - F ( y ) , x - y) >_ 0 for ull x, y • R n) or, even, if F is a P-function (maxl<_i<_n{(Fi(x) - F i ( y ) ) ( x i - Yi)} > 0) this hypothesis holds, but the reciprocal is not true. For example, the matrix:

:[00 :]

is column-sufficient, but not positive semidefinite, therefore, F is not monotone. Moreover, M is not a P - m a t r i x either, so F is not a P-function.

Finally, let us show t h a t the hypothesis of this theorem is sharp and cannot be relaxed to, say, P0-funetion. In fact, consider the following matrix

This is a P0-matrix, but F ( x ) = M x + q does not verify the hypothesis of Theorem 2, since M is not column-sufficient. Obviously, (0, 0) is a solution of the NCP. However, taking L = 2, we have t h a t all the points of the form (t, 2) for t •

[2/3, 2],

are solutions of the BVIP defined by F ( x ) = M x + q, ~ = 0, r = L. So, these points are stationary points of the associated optimization problem but, clearly, they are not solutions of the NCP.

C O N C L U S I O N S

When, for some nonlinear programming reformulation of a complementarity or variational inequality problem, it is known t h a t every stationary point is a solution, it can be conjectured t h a t standard minimization algorithms will be effective for finding a solution, since these algorithms generally find, in the limit, stationary points. However, at least from the theoretical point of view, the effectiveness of the minimization approach is not proved unless, eventually, a bounded level set can be reached. Otherwise, there could be no convergent subsequence at all. In this paper we proved t h a t a reformulation of nonlinear complementarity problems satisfies the desired requirements under weaker conditions t h a n the ones established in previous works.

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12 R. ANDREANI AND J. M. MARTfNEZ

R E F E R E N C E S

1. F. Facchinei, A. Fischer and C. Kanzow, A semismooth Newton method for variational inequalities: The case of box constraints, In Complementarity and Variational Problems: State of Art, (Edited by M. C. Ferris and J.-S. Pang), pp. 76-90, SIAM, Philadelphia, (1997).

2. M. Kojima, N. Megiddo, T. Noma and A. Yoshise, A unified approach to interior point algorithms for linear complementarity problems, In Lecture Notes in Computer Science, Volulme 538, Springer-Verlag, Berlin, (1991).

3. J.J. Mor~ and W.C. Rheinboldt, On P- and S-functions and related classes of n-dimensional nonlinear mappings, Linear Algebra and Its Applications 6, 45-68, (1973).

4. T. Wang, I~.D.C. Monteiro and J.-S. Pang, An interior point potential reduction method for constrained equations, Mathematical Programming 74, 159-195, (1996).