(Marcotte and Dussault [9]) are provided.

Transcription

1 SIAM J. CONTROL AND OPTIMIZATION Vol. 27, No. 6, pp , November Society for Industrial and Applied Mathematics 002 A SEQUENTIAL LINEAR PROGRAMMING ALGORITHM FOR SOLVING MONOTONE VARIATIONAL INEQUALITIES* PATRICE MARCOTTE? AND JEAN-PIERRE DUSSAULT$ Abstract. Applied to strongly monotone variational inequalities, Newton s algorithm achieves local quadratic convergence. In this paper it is shown how the basic Newton method can be modified to yield an algorithm whose global convergence can be guaranteed by monitoring the monotone decrease of the "gap function" associated with the variational inequality. Each iteration consists in the solution of a linear program in the space of primal-dual variables and of a linesearch. Convergence does not depend on strong monotonicity. However, under strong monotonicity and geometric stability assumptions, the set of active constraints at the solution is implicitly identified, and quadratic convergence is achieved. Key words, method mathematical programming, variational inequalities, nonlinear complementarity, Newton s AMS(MOS) subject classifications. 49D05, 49D10, 49D15, 49D35 0. Introduction. In this paper we consider the variational inequality problem defined on a convex compact polyhedron in R n. Since this problem can be formulated as a fixed-point problem involving an upper semicontinuous mapping, it can be solved by simplicial or homotopy methods for which there already exists a vast literature (see Zangwill [16], Todd [14], Saigal [13]). For large-scale problems, however, these algorithms tend to become inefficient, both in terms of computer memory and running time requirements. This explains the renewed interest in algorithms closely related to procedures originally devised for iteratively solving systems of nonlinear equations (Ortega and Rheinboldt [10]) such as the Jacobi, Gauss-Seidel, and Newton schemes (see Pang and Chan [11], Josephy [5], Robinson [12]) or projection algorithms (Bertsekas and Gafni [2], Dafermos [4]) where the cost function is approximated, at each iteration, by a simpler, e.g., linear, separable, or symmetric function. Local or global convergence of the latter methods usually hinges on the a priori knowledge of lower bounds for the Lipschitz constant of the cost function, either in a neighborhood of a solution (for local convergence) or uniformly on the feasible domain (for global convergence). These conditions are difficult, while not impossible, to verify in practice. Our approach is basically different. We choose as a merit function the complementary term (or gapfunction associated with the primal-dual formulation ofthe variational inequality and find its global minimum by application of a first-order minimization algorithm. For monotone cost functions, we show that the algorithm converges globally to an equilibrium solution and possesses the finite termination property if the function is affine. Furthermore, under geometric stability and strong monotonicity assumptions, the algorithm implicitly identifies the set of constraints that are binding at the equilibrium solution, and convergence toward the equilibrium solution is quadratic. Numerical results comparing this method to Newton s method with and without linesearch (Marcotte and Dussault [9]) are provided. * " Received bythe editors February 25, 1985; accepted for publication (in revised form) February 24, D6partement de Math6matiques, Coll6ge Militaire Royal de Saint-Jean, Richelain, Qu6bec, J0J 1R0, Canada. This research was supported by National Sciences and Engineering Research Council of Canada grants 5491 and 5789, and Academic Research Program ofthe Department of National Defense grant FUHBP. $ D6partement de Math6matiques et d informatique, Universit6 de Sherbrooke, Boul. Universit6, Sherbrooke, Qu6bec, J1K2R1, Canada. 1260

2 SEQUENTIAL LINEAR PROGRAMMING ALGORITHM Problem formulation. Notation and basic definitions. Let {Bx <-_ b}, where B is an m x n matrix (m > n), represent a nonempty convex compact polyhedron in R" and let F be a continuously, differentiable function from into R" with Jacobian F. The variational inequality problem (VIP) associated with F and consists in finding some vector x* in called an equilibrium solution, satisfying the variational inequality (VI):. () (x*-x) t:(x*) <-o for all x in Since an equilibrium solution is a fixed point of the upper semicontinuous mapping defined by x- T(x)= {arg maxya, (x-y) F(x)} it follows from Kakutani s Theorem [6] and the compactness of that the set S of equilibria is nonempty. If the Jacobian F (x) is symmetric for all x in then the function F(x) is the gradient of some function f: R", and (1) is the mathematical expression of the first-order necessary conditions corresponding to the optimization problem: (2) rain f(x) F( t) dt where the line integral is independent of the path of integration and therefore unambiguously defined. In order that a feasible point x be an equilibrium, it is necessary and sufficient that x be optimal for the linear program (3) min ytf(x). ye The optimality conditions for (3) are met by x if and only if we have A >- O, F(x) + BtA 0 dual feasibility, (4) At(Bx-b) =0 complementary slackness, Bx <- b primal feasibility. In the following, (4) will be referred to as the complementary formulation of VIP. If F is symmetric, (4) corresponds to the Kuhn-Tucker necessary optimality conditions for the optimization problem (2). If the constraint set is not polyhedral, a formulation similar to (4) can be obtained by imposing a suitable constraint qualification condition on the problem. The constraints Bx <= b will be referred to as the structural constraints associated with the variational inequality problem, and the constraints F(x)+ BtA --O, A -> 0 as the nonstructural constraints. DEFINITION 1. The function F is (i) Monotone on if (x-y)(f(x)-f(y))>=o for all x, y in ; (ii) Strictly monotone on if (x-y)(f(x) F(y))>0 for all. x, y in (x y); (iii) Strongly monotone on if there exists a positive number K such that (x-y) (F(x)-F(y))>-Kllx-yl[ 2 for all x,y in. When F is the gradient of some ditterentiable function f, then the various concepts of monotonicity previously defined correspond, respectively, to convexity, strict convexity, and strong convexity off on For ditterentiable functions, we also have the following characterization (see Auslender [1]): (i) Monotonicity on : (x-y) F (x-y)>=o for all x, y in ;, (ii) Strong monotonicity on : (x-y)tf (x)(x-y)>= llx-yll for all x, y in for some positive number r.

3 1262 P. MARCOTTE AND J.-P. DUSSAULT The solution set S of (1) is nonempty, as noted earlier, convex if F is monotone,, and a singleton if F is strictly monotone. DEFINITION 2. The gap function associated with a VIP is defined, for x in as g(x)=max(x-y) F(x). y It is clear that a feasible point x is a solution of VIP if and only if it is a global minimizer for the gap function, i.e., g(x) 0. Using this concept, VIP can be formulated as the linearly constrained optimization problem (5) min g(x). xi9 Although, in general, neither quasiconvex nor ditterentiable, it will be shown in Lemma 3 that any stationary point of (5) is an equilibrium solution. In particular, a globally convergent algorithm using the gap function as a merit function has been proposed by Marcotte [7]. DEFINITION 3. The dual gap function associated with VIP is defined as,(x)=max(x-y) F(y). ye The dual gap function is convex, but its evaluation requires the solution of a nonconvex (in contrast with linear for the gap function) mathematical program. Under a monotonicity assumption, any global minimizer of the problem minxes, (x) is a solution to VIP. A solution algorithm based on direct minimization of the dual gap function can be found in Nguyen and Dupuis [17]. DEFINITION 4. We say that VIP is geometrically stable if (y-x*) F(x*)<=0 for any equilibrium solution x* implies that y lies in the optimal face T*, i.e., the minimal face of containing the set S of all solutions to VIP. The above stability condition, especially useful when S is a singleton, ensures that T* is stable under slight perturbations to the cost function F. It is implied by the generalization to VIP of the usual strict complementarity condition: (6) {Bx* b :> a * > 0} where A* is an optimal dual vector corresponding to x* in the complementarity formulation (4). If F is strongly monotone, then geometric stability implies the strong regularity condition of Robinson 12]. Also, under geometric stability, there must exist at least one solution of VIP satisfying the strict complementarity condition (6); however it need not be unique, and there might exist optimal primal-dual couples that are not strictly complementary. Figure 1 provides examples where geometric stability holds while strict complementarity is not satisfied. In the first case, the problem is caused by a redundant constraint, while in the second case it is due to the linear dependence of the constraints gradients at x*. 2. Newton s algorithm. Since Newton s method is central to our local convergence analysis we recall its definition and main properties. Applied to VIP, Newton s method generates a sequence of iterates {x k} where x is any vector in and xk+(k >_-0) is a solution to the VIP obtained by replacing F by its first-order Taylor expansion around x k, i.e., (7) (x + -y)t(f(x )+ F (x )(x-x ))<-O ryes. The linearized problem will be denoted LVIP (x) and its (nonempty) set of solutions

4 SEQUENTIAL LINEAR PROGRAMMING ALGORITHM F(x*) redundant constraint (a) dependent constraints gradients (b) FIG. 1. Geometric stability does not imply strict complementarity. (a) Redundant constraint. (b) Dependent constraints gradients. NEW (xk). The gap function associated with LVIP (xk) will be denoted Lg (xk, x) and its mathematical expression is (8) Lg (x k, x) max (x y) (F(x k) + F (xk)(x xk)). y (the linearized gap function) In a similar fashion we define the linearized dual gap function L, (x k, x)" (9) Lp, (x k, x)=max (x-- y) (F(xk) + F (xk)(y-- xk)). y When F is strongly monotone and its F is Lipschitzian, it can be shown that Newton s method is locally quadratically convergent. We quote Pang and Chan s [11] version of this result, also obtained by Josephy [5]. THEOREM 1. If the matrix F (x*) is positive definite and the function F is Lipschitz continuous at x* then there exists a neighborhood N of x* such that if x k N then the sequence {x k} is well-defined and converges quadratically to x*, i.e., there exists a constant such that (10) lix x*ll llx x*ll Vk such that x k N where I1" denotes the Euclidian norm in R". The next result shows that Newton s algorithm has the capability of identifying T*. Actually we will prove this result for a broad class of approximation algorithms where, at each iteration, x k+l is defined as a solution to a VI where F(x) is replaced by the function G(x, x k) parameterized in x k and such that (11) (i) G(x, y) is strictly monotone in x; (12) (ii) G(x, y) is continuous as a function of (x, y); (13) (iii) G(x, x)-- F(x). Property (i) above ensures that x k+ is unambiguously defined. Property (iii) ensures that if x k+ x k then x k is the solution to the original VIP. In many practical situations, G is chosen as a strongly monotone function with symmetric Jacobian. Popular choices

5 1264 P. MARCOTTE AND J.-P. DUSSAULT for G are" Gi(x, y) Fi(Yl,, Yi-1, xi, Yi+l, Y,,), 1,., n Jacobi iteration, Gi(x, y) Fi(x1, Xi, Yi+l, Yn), 1,, n Gauss-Seidel iteration, G(x, y)- F(y)+ F (y)(x- y) Newton s method, G(x, y) Ax + p[f(y) Ay] Projection method, where p > 0 and A is a symmetric positive definite matrix. Other choices for G may be found in Pang and Chan [11] and Marcotte [8]. PROPOSITION 1. Assume that F is monotone and that geometric stability holds for VIP. Let X k+l be a solution to the VI: (xk+i y)tg(xk+l, xk)<0= for all y e where G satisfies (11), (12), (13). Then, for each optimal solution x* of VIP there exists a neighborhood V of x* such that if x k V then x k+l T*. Proof Assume that the result does not hold. Then there exists an extreme point u of-t* and a subsequence {Xk}k converging to some x* such that (x k+l u)tg(xk+, x k) 0 for all k e I. Taking the limit as k--> o (k e I) we obtain implying, by geometric stability, that u (x* u)tf(x *) (x*- u)tg(x *, x*) <= 0 T*, a contradiction. 3. A linear approximation algorithm. In this section we present a model algorithm for solving VIP based on its complementarity formulation (4) that proceeds by successive linear approximations of both the objective and the nonstructural, usually nonlinear, constraints. Throughout this section the function F will be assumed monotone with Lipschitz continuous Jacobian F. Any solution to (4) is clearly a global minimizer for the following (usually) nonconvex, nonlinearly constrained mathematical program" def min h(x, A) ht(b-bx)=xtf(x)+bth x,a (14) subject to F(x)+Bth=0, Bx<-b, h>-o. The following lemma relates the objective in (14) to the gap function. LEMMA 1. We have (15) Proof. g(x)=minh(x,h) subject to F x + B ta O, A >- O. g(x) max (x y) F(x) ye x F(x)- min ytf(x) By<--b =xf(x)- max b Brl F( la,o

6 SEQUENTIAL LINEAR PROGRAMMING ALGORITHM 1265 by linear programming duality theory. Hence g(x)=x F(x)+ min b A F(x)q-BtA =0 Z--->0 after setting A -/x, and the result follows if we replace F(x) by the equivalent term -BtA. The next lemma, basic to our global convergence analysis, states that any stationary point of the mathematical program (14) is actually an equilibrium solution to VIP and justifies the use of an algorithm based on identifying points satisfying first-order conditions of (14). The proof does not rely on any sort of constraint qualification for the nonlinearly constrained problem (14). LEMMA 2. Let (2,.) be a vector satisfying the first-order necessary optimality conditions for (14). Then is a solution to VIP. Proof. It suffices to show that h(, )= 0. Assume that h(, )> 0. Without loss of generality we also assume that h(, )= g(); otherwise, for the linear program (16) min h(x, A) A subject to F(:)+B A=0, A>-0 would not be optimal and an optimal A-solution to (16) would constitute, together with 2, an obvious descent direction for h at (2, ). Consider the linearized problem LVIP (2) with its gap function Lg (:) and complementarity formulation: (17) min h(x, A) def xt[f(,)h F (g)(x-g)] + bra subject to F(g) + F ()(x- 2) + B A O, A=>0. Problem (17) constitutes a positive semidefinite quadratic program whose optimal solution s primal vector corresponds to a (not necessarily unique) Newton direction. Consider a Frank-Wolfe direction d (Y 2, ]) for (17) at the point (2, ). Direction d is a feasible descent direction for the linearized gap function Lg at 2. Since V h(g,,) is identical with N. V/(:, ), and so are the directional derivatives of Lg and g, it follows that-is also a feasible descent direction for g at We are now in a position to give a precise statement of our algorithm. ALGORITHM Initialization. Let x be any vector in and and set k <-- 1. while convergence criterion not met do 1) Find descent direction d. A o arg min b A subject to F(x) + B A 0

7 1266 P. MARCOTTE AND J.-P. DUSSAULT (18) Let (dx(xk), d(xk)) be an extremal solution to the linear problem min xt(f(xk)+ F"(xk)xk)+ b A AO subject to F(x k) + F (xk)(x x k) + B A O. (G(xk)- x Set d 2) Perform arc search on the gap function. (19) if g(d(xk))<=1/2g(x k) then ff-i else 3) Update. endwhile. Some comments are in order: x+,,_ x ffarg min g[xk+o(dx(xk)--xk)]. 0[0,1] + g(d(x)-x) A k+ earg min F(xk+I)+BtA =0 k-k+l (1) At step 2) of Algorithm N, the minimization, with respect to the primal vector x, of the nonditterentiable objective g could be seen as a search along an arc in the space of primal-dual variables (x, h). Since dual vectors h have to be computed repeatedly, this operation can be carried out efficiently using reoptimization techniques of linear programming. (2) It is not required, or even advisable, that the arc search be carried out exactly. For instance, the Armijo-Goldstein stepsize rule, or any rule guaranteeing a "sufficient" decrease of the objective along the search direction could be implemented. (3) For affine functions F Ax + a, Algorithm N reduces to the standard Frank- Wolfe procedure for solving quadratic programming problems, as then the nonstructural constraints become linear. N. 4. Convergence analysis. We first state and prove a global convergence result for Algorithm PROPOSITION 2. Any point of accumulation of a sequence generated by Algorithm N is an equilibrium solution. Proof. If g(dx(xk))--<.5g(x k) infinitely often at (19), then limk_o g(xk) =0. Otherwise the linesearch in (19) is asymptotically always performed and, to prove global convergence, we will strive to check the conditions behind Zangwill s global convergence theorem, namely: (i) All points generated by the algorithm lie in a compact set. (ii) The algorithmic map is closed outside the set of solution points S. (iii) At each iteration, strict decrease of the objective function occurs. (i) Since is compact by assumption, it is sufficient to show that the sequence {dx (xk)} is bounded. By definition of the sequence {d(xk)} we have (20) d, (x k) arg min b A AO subject to B A H(x k) where H(xk) r F(xk)+ F,(xk)(dx(Xk)_xk) R".

8 SEQUENTIAL LINEAR PROGRAMMING ALGORITHM 1267 (21) First observe that the linear program: min bta A0 subject to B A -H(x k) is the dual of the linear program (22) max -xf(x). that is feasible and bounded; hence, by linear programming duality, we have that (17) is also feasible and bounded, i.e., that (17) possesses at least one optimal basic solution. Let {Ne}e=,...,p denote the set of full rank square submatrices (basis) of B Since da (x) is extremal, we have d (x) -N-H(x) for some e e (1,, p}. From the continuity of F and F we deduce that H(x) must lie in some compact set K independent of x k. Therefore d(x)e C der UP=_- -N-K, which is bounded. The same continuity argument is then used to show boundedness of the sequence {+}. (ii) The closedness of the algorithmic map follows directly from the continuity of F and the closedness of the linesearch strategy used. (iii) We must prove that h(x+, +)< h(x,,) if the latter term is positive (not zero). This is a direct consequence of Lemma 2. [3 PROPOSITION 3. If F is monotone on and affine, then Algorithm N * converges in a finite number of iterations. Proof Replacing F by Ax + a in (16) yields a quadratic programming problem. Its solution set is a face T of the polyhedron {Ax + B & O, Bx <= b,, >- 0}. For some iterate k we must have that (dx(x), d(x)) lies in (otherwise the iterates would always be bounded away from T, When (dx(x), da(xk))e ". = we have contradicting global convergence of the method). 1 and (xk+l, k+l) [-] Remark. The preceding result is also valid under the assumption that T* is a singleton (F monotone but not necessarily affine). The proof is similar. To obtain a rate-of-convergence result for Algorithm N we assume, until explicitly stated otherwise, that the function F is strongly monotone in a neighborhood of the solution x* with strong monotonicity coefficient and that the geometric stability condition is satisfied at x*. This implies that the entire sequence {x} converges to the unique solution x*. Under these assumptions we will show that Algorithm N * is locally equivalent to Newton s method, thus implying quadratic convergence and implicit identification of the set of active constraints at x*. We first show that the descent direction d obtained from Algorithm N * satisfies d),(x) NEW (xk) if x is sufficiently close to x*. The following lemmas will be used in the proof. LEMMA 3. The optimal dual vector y(x) associated with the nonstructura! constraint F(x)+ B, =0 of (18) satisfies lim_ y(x) x*. Proof Write the Lagrangian dual of the linear program (18)" max min x [F(xk) + F t(x)x] + b A y[ F(x) + F (x)(x-xk) + B, ]. y xccd Then observe that the inner minimum has value -oe unless By <= b, in which case the minimum over nonnegative A is achieved when I is zero, yielding max min xt[ F(x) + F" (x)x y [ F(x) + F (xk)(x x )]. yc xc

9 1268 P. MARCOTTE AND J.-P. DUSSAULT This expression is equivalent, modulo a constant term, to (23) max min (x--y)t[f(xk)nt-f (xk)(x--xk)]--(x--xk)f (xk)(x--x k) ycb x and constitutes a quadratic perturbation of the dual gap function L at x k. Since y(x k) is dual-optimal for (18) it must correspond to the y-part of a solution to (23). If y(xk) does not converge to x* then there exists a subsequence {Xk}k1 such that limk_oo.ky(xk) 37 X*. Passing to the limit in (23) we obtain, after setting x to : lim max min (x--y) [F(xk)+ F (Xk)(x--xk)]--(x--xk) F (xk)(x--xk) kcx y x ki But this taking y (24) _<_ (7-7) [F(x*) + V (x*)(;- x*)] _-<-KJJ)7-x*ll 2 <0. by strong monotonicity (- x*) F (x*)(- x*) contradicts the optimality of the sequence {y(xk)}k1 since we obtain, by x*" lim min (x--x*) [F(xk)+ F (Xk)(X--Xk)]--(X--Xk) F (xk)(x--x k) k-c x min (x x*) F(x*) =0 by definition of x*. LEMMA 4. There exists an index K such that k >-K implies d,,(x) T*. Proof From (23) we get dx(xk)e D(x k) dez arg min (x-- y(xk)) [F(xk)+ F (xk)(x--xk)] (x x) F (x)(x x). Since y(xk)-+ X* as koo (Lemma 3), (24) represents, for x k close to x*, a small quadratic perturbation of the linear program: min,a, x F(x*). It follows from the geometric stability assumption that dx(x k) T*. [3 COROLLARY. dx(x*) T*. Proof Since F is continuous, the point-to-set mapping :-+{d,,(ff)} is upper semicontinuous. Hence dx(x*){dx(limk_,ooxk)}=limk_,oo{dx(xk)} T*. V] LEMMA 5. limk+oo d,,(xk) x*. Proof From the proof of Lemma 2, we have g (xk; dx(xk)--xk)<o. Passing to the limit and using upper semicontinuity there comes g (x*; dx(x*) x*) <= O. But, by Danskin s rule of differentiation of max-functions (see [18]), we have. g (x*" dx(x*)-x*)=max [dx(x*)-x*] [F(x*)- F"(x*)(y-x*)]. yet* Assume that T* is not the singleton x* (otherwise the result follows trivially from Lemma 5) and let e be a positive number such that o-rx*-e(d(x*)-x*) T* (see

10 Fig. 2). Then we have SEQUENTIAL LINEAR PROGRAMMING ALGORITHM 1269 o->_ g (x*; d(x*)-x*), >- [dx(x*)-x*] [(x*)+ F"(x*)(d(x*)-x*)] >=,lldx(x*)-x*ll implying that &, (x*) x*. FG. 2 LEMMA 6. There exists an index K such that for k >= K, dx(x k) NEW (xk). Proof From (18), d (x k) is an optimal dual vector for the linear program (25) min z [F(x )+ F (xk)(dx(xk)--xk)]. For k large, d(x k) is close to x* (Lemma 5) and problem (25) is an arbitrary small perturbation of the linear program min z F(x*) T*, by definition. Therefore the optimal solutions to whose set of optimal solutions is (25) lie in T* by geometric stability. From the complementary slackness theorem of linear programming we can write d, (xk)t( Bd(xk) b O. We conclude that the couple (d,(xk), d(xk)) is optimal for the quadratic program (17). Since its solution is unique in x and equal by definition to NEW (xk) we conclude that d(x k) NEW (xk). ] PROPOSITION 4. There exist positive constants and such that IIx-x*ll g(x) llx- x*ll,

11 1270 P. MARCOTTE AND J.-P. DUSSAULT Proof It suffices to prove the result in a neighborhood of x*. (26) (i) Proof that g(x) -<_/3 IIx x*[[. g(x)=max(x-y) F(x) (x x*) F(x) + max (x* y)tf(x) <-IIx-x*ll. IIF(x)ll +max (x*- y) F(x*) ycd +max (x*- y) (F(x). diameter of Then set M + M D. (27) (ii) Proof that g(x) >_- a ][x- F(x*)) --< IIx- x*ll, IIV(x)ll / DIIx- x*ll sup <= M + M diam where M d=supxa, ]]F(x)II, M d--esup,,a,,, ]lf(()-f(rl)[i/i]-rl]l, and D is the We consider three mutually exclusive cases. Case 1. T*= x*. (Fig. 3.) For x sufficiently close to x* we have g(x)=(x-x*) V(x) (x x*) F(x*)+ (x x*) (f(x) F(x*)) => llx-x*ll, IIF(x*)ll cos (x-x*, F(x*))/,, IIx-x*ll 2. F(x*) FIG. Case 1. Since F(x*) is orthogonal to no feasible direction from x* (by geometric stability) we must have that cos (x-x*, F(x*)) is positive and bounded away from zero. Hence (27) holds with def a c inf {cos (- x*, F(x*))}llF(x*)ll > o. :x*

12 SEQUENTIAL LINEAR PROGRAMMING ALGORITHM 1271 Case 2. T* {x*} and x T*. (Fig. 4.) Let p be a positive number such that the mapping Proj. (x-(1/p)f(x)), defined for x cp, is contracting, where Proj. denotes the projection operator on P, in the usual Euclidean norm. The existence of such a number p is a consequence of, say, Example 3.1 of Dafermos [4]. For x sufficiently x P x* Y -F(x*) x- f(x) FIG. 4. Case 2. [0, 1] be close to x*, p a er Proj. (x-(1/p)f(x)) lies in T* (see Proposition 1). Let 0 the contraction constant, dependent on p; we have lip- x*ll <--ollx-x*ll. we have (28) Also by construction of p IIp-xll>= [Ix-x*]l-[Ix*-pl[ >-_(1-o)llx-x*ll. (29) (x-p) F(x)- Ilx-p 2. Define (30) 0=max {4)lx+ck(p-x) r*} by the triangle inequality (0 must be positive since x lies in the relative interior ri (T*)) and y=x+o(p-x). We have (x- y)tf(x) q,(x-p) F(x) =Ollx-pll = by (29) pllx-pll(1-o)llx-x*ll by (28). Now 4,1lx-pll- Ilx-y[I must be bounded from below by some positive number s since x lies in ri (T*) and )7 is on the boundary of T*. It follows that g(x)>=(x-y) F(x) p(1-o)sllx-x*ll and the result holds with ce ps(1-0). Case 3. x : T* (consequently T* - ). (Fig. 5.) Define p Projr. (x). First we will show that cos (x-p, F(x*)) is bounded below by some positive number % Define, for x T*,, the function x r/(x) where r/(x) is the intersection of the line going through the segment [p, x] with the boundary of in the direction x-p. Let Be(x*) be a ball of radius e about x*, H B(x*)f3 -T*, and E the closure of r/(h) (see Fig. 6). We have E f l T*= and (31) cos (x-p, (x*)) cos (n(x)-p, F(x*)) ->min cos (v-p, F(x*)).

13 1272 e. MARCOTTE AND J.-P. DUSSAULT X T* x* x-f(x) FIG. 5. Case 3. F(x*) E FIG. 6 But cos(v-p,f(x*))>o (pet*) for each vc_t* by geometric stability. Hence, cos (x-p, F(x*)) >= 3/ > O. We then write (x-p)tf(x*) >- llx-pll" IIF(x*)ll. Thus >3 (32) (x-p)tf(x)=-[ix-pll IIF(x*)ll for x sufficiently close to x*. Now consider the following two subcases. Case 3.1. IIx-pll<-_llp-x*ll with C=ps(1-O)/2ODM. Define 37 as in Case 2 (see Fig. 4). Then g(x) >= (x-.9)tf(x) (x -p) F(x)+ (p -y) F(x) O+ (p -.9) F(p) + (p y) (F(x) F(p)) >-os(a-o)llx-x*ll-rllp-x*[idm since pc T* (see Case 2) _-> (ts(1 o) ODM )IIx x*ll >ps(1-o) 2 Set c ps 1 O) / 2. Case 3.2. IIx-Pll--> ffllp-x*l[. We have (33) [Ix-x*ll Ilx-pll + p-x*ll (1 + ff)iix-pll.

14 SEQUENTIAL LINEAR PROGRAMMING ALGORITHM 1273 We obtain g(x)>=(x-p) F(x) llx-pl[" F(x*) by (32) -2 >_..y 1 x* =2 1+ [Ix- I1" IIF( x* by(28)and(29),withf(x*)o, andtheresultholdswitha= yllf(x*)ll/2(l+). [3 Remark 1. The above general proof does not require ditterentiability of the cost mapping F. If F is ditterentiable, the proof of Proposition 4 can be somewhat streamlined (see Dussault and Marcotte [21]). Remark 2. Proposition 4 strengthens a result of Pang 19] who derives an estimate of the form IIx- x*ll oo4g(x) N. for some positive constant w. PROPOSITION 5. Let {x k} be a sequence generated by Algorithm Then there exists an index K such that for k >-K, x k+l= NEW (xk), the Newton iterate. Proof We must prove that g(new(xk))<-1/2g(x) for k>=k, in which case Algorithm N will set x +1 to d,(x), which is equal to NEW (x) by Lemma 6: g(new(x))=</3llnew(x)-x*ii by Proposition 4 <- t3cllx-x*ll 2 from (10) _<-- c ilxk _x, llg(x) by Proposition 4 1 <_-g(x) 2 If as soon as IIx-x*ll c/2c, l-1 The preceding results can be summarized in a theorem. THEOREM 2. Consider a VIP with monotone cost function F and let {x} be a sequence generated by Algorithm N. (i) g(x k+l) < g(x k) if g(x) O. (ii) lim_. g(x) O. VIP is geometrically stable, (iii) IfF is affine or T* is a singleton then there exists an index K such that g(x) 0 for k >-_ K (finite convergence). (iv) If F is strongly monotone then the sequence {x} converges quadratically to the point x* and there exists an index K such that x T* whenever k >= K. 5. Numerical results. A working version of Algorithm N has been developed, using a standard linear programming code, and contrasted against Newton s method, with or without linesearch. The asymmetric linear complementarity subproblems in Newton s method have been solved by Lemke s Complementary Pivoting Algorithm. then

15 1274 P. MARCOTTE AND J.-P. DUSSAULT In the test problems, has been taken as the unit simplex i=1 Xi--- 1, x >= 0 and the mapping F assumed the general form F(x) (A- A )x + B Bx + yc(x) + b where the entries of matrices A and B are randomly generated uniform variates, C(x) is a nonlinear diagonal mapping with components Ci(x) arctan (xi), and the constant vector b is chosen such that the exact optimum be known a priori. The parameter y is used to vary the asymmetry and nonlinearity of the cost function. Sixteen five-dimensional and sixteen 15-dimensional problems have been generated, with y-values ranging from Newton s search direction differs from Algorithm N s direction in 18 of the 32 problems. In some instances (Figs. 7-10) Algorithm N yields a direction as good or better than Newton s direction. For some other problems (Figs ) Newton s direction is slightly superior. In all cases, the difference in the number of iterations required to achieve a very low gap value is small. 1.4 Dimension 15, <(C) ALG.N Iteration k NEWTON o-o-o- NEWTON (with step) FIG Dimension- 15, y ALG.N # Iteration k NEWTON o-o-o- NEWTON (with step) FG. 8

16 SEQUENTIAL LINEAR PROGRAMMING ALGORITHM Dimension 5, y o Iteration k ALG.N NEWTON o-o-o-o- NEWTON(with step) FIG Dimension 5, y Iteration k ALG.N NEWTON NEWTON(with step) FIG Dimension =15,7 = , Iteration k NEWTON o--o-o- FIG NEWTON(with step)

17 1276 P. MARCOTTE AND J.-P. DUSSAULT Dimension 5, " Iteration k ALG.N # NEWTON o-o-o-o- NEWTON(with step) c3-tzc3- FIG, 12 t3.8 Dimension 15, Iteration k ALG.N u t, NEWTON c)cc)c) NEWTON(with step) [] [] [] FIG o0 Dimension 5, <(, 0.6 N o " Iteration k ALG.N NEWTONo-o-o-o- NEWTON(with step) FIG. 14

18 SEQUENTIAL LINEAR PROGRAMMING ALGORITHM 1277 This preliminary testing shows some promise for the linearization algorithm. Its direction finding subproblem involves a linear program, versus an asymmetric linear complementarity problem for Newton s method. The linear subproblem bears close resemblance to the linear program that must be solved to evaluate the gap function, and as such could benefit from some fine tuning of the computer code. Moreover, it may well prove unnecessary to solve the subproblem exactly, yielding another area for further improvement. In contrast, solving linear complementarity problems yields a feasible solution only at termination, therefore making the implementation of an inexact strategy more difficult. Finally let us mention that Marcotte and Gu61at [20] have successfully implemented Algorithm N to solve large-scale network equilibrium problems when the mapping F, i.e., its Jacobian matrix, is highly asymmetric. 6. Conclusion. The main result of this paper has been to prove global and quadratic convergence of an algorithm for solving monotone variational inequalities. The algorithm operates by solving linear programs in the space of primal-dual variables. Computational experiments show that the algorithm is efficient for solving both smallscale and large-scale problems. Acknowledgments. The authors are indebted to anonymous referees for relevant comments on an earlier version of this paper that led to numerous improvements. REFERENCES [1] A. AUSLENDER, Optimisation. Mdthodes numdriques, Masson, Paris, [2] D. BERTSEKAS AND E. M. GAFNI, Projection methods for variational inequalities with application to the traffic assignment problem, Math. Programming Stud., 17 (1982), pp [3] R.W. COTTLE AND G. B. DANTZIG, Positive (semi) definite programming, in Nonlinear Programming, J. Abadie, ed., North-Holland, Amsterdam, [4] S. C. DAFERMOS, An iterative scheme for variational inequalities, Math. Programming, 26 (1983), pp [5] N. H. JOSEPHY, Newton s method for generalized equations, Technical Report 1966, Mathematical Research Center, University of Wisconsin, Madison, WI, [6] S. KAKUTANI, A generalization ofbrouwer sfixed point theorem, Duke Math. J., 8 (1941), pp [7] P. MARCOTTE, A new algorithm for solving variational inequalities, with application to the traffic assignment problem, Math. Programming, 33 (1985), pp [8] Algorithms for the network oligopoly problem, J. Oper. Res. Soc., 38 (1987), pp [9] P. MARCOTTE AND J.-P.fDussAULT, A note on a globally convergent method for solving monotone variational inequalities, Oper. Res. Lett., 6 (1987), pp ] J. M. ORTEGA AND W. C. RHEINBOLDT, Iterative Solution ofnonlinear Equations in Several Variables, Academic Press, New York, [11] J. S. PANG AND D. CHAN, Iterative methods for variational and complementarity problems, Math. Programming, 24 (1982), pp [12] S. M. ROBINSON, Generalized equations, in Mathematical Programming: The State of the Art, A. Bachem, M. Gr6tschel, and B. Korte, eds., Springer-Verlag, Berlin, New York, 1983, pp [13] R. SAIGAL, Fixed point computing methods, in Operations Research Support Methodology, A. G. Holzman, ed., Marcel Dekker, New York, ] M.J. TODD, The Computation offixed Point and Applications, Springer-Verlag, Berlin, New York, [15] W. I. ZANGWILL, Nonlinear Programming: A Unified Approach, Prentice-Hall, Englewood Cliffs, NJ, [16] W. I. ZANGWILL AND C. B. GARCIA, Equilibrium programming: the path-following approach and dynamics, Math. Programming, 21 (1981), pp [17] S. NGUYEN AND C. DUPUIS, An efficient method for computing traffic equilibria in networks with asymmetric transportation costs, Transportation Sci., 18 (1984), pp [18] J. M. DANSKIN, The theory ofmax-min, with applications, SIAM J. Appl. Math., 14 (1966), pp

19 1278 P. MARCOTTE AND J.-P. DUSSAULT [19] J. S. PANG, A posteriori error bounds for the linearly-constrained variational inequality problem, Math. Oper. Res., 12 (1987), pp [20] P. MARCOTTE AND J. GUELAT, Adaptation of a modified Newton method for solving the asymmetric traffic equilibrium problem, Transportation Sci., 22 (1988), pp [21] J.-P. DUSSAULT AND P. MARCOTTE, Conditions de r.gularitd gom.trique pour les indquations variationnelles, RAIRO Rech. Op6r., 23 (1988), pp