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JOURNAL OF OPTIMIZATIONTHEORY AND APPLICATIONS: Vol. 44, No. 3, NOVEMBER 1984

A Descent Algorithm for Solving Monotone Variational Inequalities and Monotone Complementarity Problems 1 M. J. SMITH2 Communicated by O. L. Mangasarian

The paper provides a descent algorithm for solving certain monotone variational inequalities and shows how this algorithm may be used for solving certain monotone complementarity problems. Convergence is proved under natural monotonicity and smoothness conditions; neither symmetry nor strict monotonicity is required.

Abstract.

Key Words. Variational inequalities, complementarity problems, traffic equilibria, monotonicity, descent algorithm.

1. Introduction

The scalar or inner product of x, y e R k will be written xy. Let D C Nk and f : D -->R k be any function. The condition

xcD

and

f(x)(y-x)~O,

forallyrD,

(1)

is called a variational inequality, tt was pointed out by Karamardian (Ref. I) that, for a function f : R k+__>Rk, the complementarity condition

x >i O,

f ( x ) >I O,

x f ( x ) = 0,

(2)

may be expressed in the form (1) by simply putting D - R ÷. This paper addresses the problem of finding an x ~ R k satisfying (1) when (i) D is the convex hull of a finite set of points in Nk and (ii) f is differentiable and has a continuous positive semidefinite Jacobian f'. _

Remark 1.1.

k

A function with a positive semidefinite Jacobian is called

monotone.

~The author is grateful to two anonymous referees for their very valuable comments on an earlier draft of this paper. 2 Lecturer, Department of Mathematics, University of York, Heslington, York, England.

485 0022-3239/84/1 I00-0485503,50/0 ~ 1984Plenum PublishingCorporation

486

JOTA: VOL. 44, NO. 3, NOVEMBER 1984

An iterative algorithm is given which converges to the set of solutions of (1) when (i) and (ii) hold. The algorithm does not require the calculation off'. We show how to apply the algorithm to solve the nonlinear complementarity problem (2) when (ii) is satisfied and, in place of (i), an R > 0 is known such that (2) has a solution x* with max ]xij < R.

l~i~n

(3)

2. Context

Variational inequalities were initially considered in Banach spaces; see Stampacchia (Ref. 2), Hartman and Stampacchia (Ref. 3), Browder (Refs. 4 and 5), Lions and Stampacchia (Ref. 6), and the recent book by Kinderlehrer and Stampacchia (Ref. 7). Variational inequalities of the form (1), with D a convex polyhedron, arise in the theory of traffic equilibrium; see Smith (Ref. 8), Dafermos (Ref. 9), or Bertsekas and Gafni (Ref. 10). They also arise in optimization. Several iterative algorithms have been proposed for the solution of nonlinear variational inequalities in a more general setting; see, for example, Glowinski (Ref. 11), Glowinski, Lions, and Tr~molie~es (Ref. 12), or Sibony (Ref. 13). On the other hand, iterative algorithms for solving the linear complementarity problem have been given by Mangasarian (Ref. 14) and Ahn (Ref. 15). Cottle (Ref. 16) provides a useful review of the complementarity problem as it stood in t977. The iterative algorithms proposed often require (in •k) that f ' be symmetric, af~l axj = ~fs/ox,, or strictly monotone. Symmetry leads to descent algorithms, as f is then a gradient, and strict monotonicity leads to contraction methods, as then the vector field - f has a contracting character. In this paper, we introduce, for variational inequalities of our special type, a descent algorithm which utilizes just the monotonicity and smoothness o f f ; neither symmetry nor strict monotonicity is required. The descent character of the algorithm allows a natural rule to be adopted for roughly optimizing the steplengths chosen. 3. Preliminaries Notation. Let II" n denote the Euclidean norm in R k, and let 0 denote the zero vector in R k.


487

Differentiability.

Let D be a closed convex subset of Nk, and let f : D-> ~k be any function. We shall say that f is differentiable if and only if there is a function f ' : D-~ Nk2 such that for, any x ~ D, x + h ~ D,

H[f(x+Ah)-f(x)]/A -f'(x)h[i->0,

as 3. ~ 0 + .

Similarly, we shall say that V: D -> ~ is differentiable if and only if there is a function V': D -> Nk such that, for any x E D, x + h c D, [V(x+Ah)-

V(x)]/~-hV'(x)--,O,

as A -~0+.

4. A Stable Dynamical System Let

u:D ~ Ek be defined by putting

u(x)=-f(x),

for a l l x e D.

Then, (1) may be written as

xeD

and

u(x)(y-x)O,

x(0) ~ D,

(9)

if - u is monotone and continuously differentiable. Lemma 5.1.

Let f =

-u

be differentiable and monotone on D. Then,

V;(x) 62(x) a . >1 - 3 / 2 , put x.+~ = x . + A.6. and 7t.+~ = A. ; if-3/2>

an, put x.+~ = x . +A.6n and An+l =2A..

Finally, put A.+I = min(A.+l, A(x.+l)). Stop if V ( x . + 2~.6.) = O.

Then, we have the following result, which is proved in the Appendix.


491

Theorem 6.1. Let f = - u be continuously differentiable and m o n o t o n e on D, and let {x,,} be an infinite sequence generated by the above algorithm. Then, the Euclidean distance d(x,, E) between x, and the set of solutions to (1) tends to zero. 7. Complementarity Problem The complementarity problem (2) m a y be written as

u(x) is normal, at x, to N~.

(10)

Let IR = [0, R], and consider the condition

u(x) is normal, at x, to I~,

(It)

Now, suppose that (10) has a solution x* with

tlx*ll +ee,

as

Ilxll- +oc,

(12)

Under these circumstances, if R is chosen so that

xf(x)>O,

for ]lxll ~> R,

then R will be appropriate. When a condition like (12) is not satisfied, the prior calculation of an appropriate R is more difficult. To deal with the general case, it would therefore be natural to extend our algorithm so as to include a method o f systematically increasing R when a current value is too small. In spite of some effort, the author has been unable to devise a way of doing this which is likely to be efficient and for which convergence can be proven. Existence of a Solution. There may, of course, be no solution to (2), even if f = - u is smooth and monotone. For example, if f(x) = (-1,-1,...,-1),

for x e Nk+,

then f is smooth and monotone and (2) has no solution.

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8. Conclusions The p a p e r gives an objective function, which measures the extent to which a current point fails to satisfy a certain type of variational inequality, and an algorithm which steadily reduces the objective function to zero in the presence of monotonicity. We have also shown how to apply the algorithm to solve certain m o n o t o n e complementarity problems.

9. Appendix Here, we give proofs of L e m m a 5,1 and Theorem 6.1.

Notation.

As previously,

x + = m a x { 0 , x}

and

x+2 = (x+) 2.

The transpose of the Jacobian of u, evaluated at x, will be written

J= J(x). We put u P ( x ) -- wlP - x,

for x ~ D,

and let

J i - Yp(x) -

b e the Jacobian of u p at x. The zero vector in R k, is to be Op.

Proof of Lemma 5.1. a J r = a p,

For any row vector a ~ R k,

say,

is the projection of - a onto

{o~} x{o~} x . . . x{o~_,} x~. x{o~+1} x . . . x{o,}. It is clear that ½v'(x) = Z [u(x)(wp -

,i

x)]+{(wp- x)J + u'(x)}

= ,~J + E u ~ ( x ) [ u ( x ) ( w f pi

x)]+.


493

Hence,

½V'(x)a(x) = ( ~ I ) ~ + E [ u ( w f - x ) ] + [ u ( w f - x)l+Eup(wf - x)] pU < - Z [ u ( w f - x ) ] + [ u ( w 7 - x)]~+ = - w ( x ) , p0

since f is monotone, and so (aJ)a co.

Further, since { V(xn)} never increases, there exists e > 0 such that

V(x,) ~ e,

for all n.

Thus, if D, = {x; x e D and V(x) ~ E}, it follows from our supposition that xn ~ D,,

for all n.

For each x ~ D - E and each A > 0, let

a(x, ;t ) = [V(x + ~ a ( x ) ) - V(x)]/a W(x). By the mean-value theorem, there exists a n u m b e r r/such that 0 < ~7 < 1 and

(x, A) = a(x) V'(x + n ~ ( x ) ) / W(x). For each x, A, let ~?(x, A) be one such ~7, and let

~(x, :~)= V'(x +n(x, ;~);t~(x)). Now,/~ is continuous, and therefore bounded, on D. Also, V' is continuous, and therefore uniformly continuous, on D. Hence, /3(x, A ) ~ V'(x),

as A -->0+,

(13)

uniformly in D. Furthermore, W is b o u n d e d below on D, (and ~ is bounded

494


on D~). Hence,

a(x,A)~6(x)V'(x)/W(x),

asA~0+,

uniformly in D~. Now, f is monotone; and so, by Lemma 5.1,

~(x) V'(x)/W(x) ~ -2, for each x ~ D - E . Therefore, by the uniform convergence of/3 in (13), there is Ao = Ao(E) such that

o~(x,~)