c 1998 Society for Industrial and Applied Mathematics

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1 SIAM J. OPTIM. Vol. 9, No. 1, pp c 1998 Society for Industrial and Applied Mathematics WEAK SHARP SOLUTIONS OF VARIATIONAL INEQUALITIES PATRICE MARCOTTE AND DAOLI ZHU Abstract. In this work we give sufficient conditions for the finite convergence of descent algorithms for solving variational inequalities involving generalized monotone mappings. Key words. sharp solution, variational inequality, descent algorithm, generalized monotonicity AMS subject classifications. 90C33, 49M99 PII. S Introduction. Recently, Burke and Ferris [5] introduced sufficient conditions for the finite identification, by iterative algorithms, of local minima associated with mathematical programs. To this aim, they introduced the notion of a weak sharp minimum, which extends the notion of a sharp or strongly unique minimum to mathematical programs admitting nonisolated local minima. In our work, we extend their results and those of Al-Khayyal and Kyparisis [1] to generalized monotone variational inequalities and provide a characterization of their solution sets. Our work is also closely related to that of Patriksson [14], who analyzed the finite convergence of approximation algorithms for solving monotone variational inequalities under a sharpness assumption. The paper is organized as follows. Section 2 introduces the main definitions. In section 3, we reformulate the variational inequality problem (VIP) as a convex program and show that its objective is continuously differentiable at any solution of the VIP, under a regularity assumption. In section 4 we introduce the notion of weak sharpness for the VIP and derive a necessary and sufficient condition for a solution set to be weakly sharp. Finally, section 5 addresses the finite convergence of iterative algorithms for solving variational inequalities whose solution set is weakly sharp. 2. Notation and definitions. Let X denote a nonempty, closed, and convex subset of R n and let F be a mapping from X into R n. We consider the VIP that consists of finding a vector x X that satisfies the variational inequality: (2.1) F (x ),x x 0 x X, where x, y denotes the Euclidean inner product of two vectors in R n. Throughout the paper, we will denote by X the set of solutions of the variational inequality (2.1). If X is a subset of R n, its polar set X is defined as X := {y R n : y, x 0 x X}. Received by the editors September 27, 1996; accepted for publication (in revised form) September 2, 1997; published electronically December 2, This research was supported by NSERC (Canada) and FCAR (Québec). DIRO, Université de Montréal, C.P. 6128, succursale Centre-Ville, Montréal, Québec, H3C 3J7 Canada (marcotte@iro.umontreal.ca). CRT, Université de Montréal, C.P. 6128, succursale Centre-Ville, Montréal, Québec, H3C 3J7 Canada (daoli@crt.umontreal.ca). 179

2 180 PATRICE MARCOTTE AND DAOLI ZHU We denote by int(c) the interior of a set C. The projection of a point x R n onto the set X is defined as proj X (x) := arg min x y. y X If X is a convex set, its normal cone at x is {y R n : y, z x 0 z X} if x X, (2.2) N X (x) := otherwise, and its tangent cone at x is T X (x) :=[N X (x)]. Using this notation, a vector x is a solution of the VIP if and only if (2.3) F (x ) N X (x ) or, equivalently, (2.4) proj TX (x )( F (x ))=0. A mapping F from a convex set X into R n is monotone on X if, x, y in X, (2.5) F (y) F (x),y x 0. It is strongly monotone on X if there exists a positive number α such that, x, y in X, (2.6) F (y) F (x),y x α y x 2. It is pseudomonotone on X if, x, y in X, (2.7) F (x),y x 0 F (y),y x 0. It is strongly pseudomonotone on X if there exists a positive number β such that, x, y in X, (2.8) F (x),y x 0 F (y),y x α y x 2. It is monotone + on C if F is monotone and for every pair of points x, y in X, (2.9) F (y) F (x),y x =0 F (y) =F (x). It is pseudomonotone + on X if F is pseudomonotone and, x, y in X, (2.10) F (x),y x 0 and F (y),y x =0 F (y) =F (x). It is quasimonotone on X if, x, y in X, (2.11) F (x),y x > 0 F (y),y x 0. Several results concerning mappings satisfying the above monotonicity or generalized monotonicity conditions can be found in Schaible [15] and Zhu and Marcotte [11, 17]. Finally, a mapping F from the set X into R n is Lispchitz continuous on X, with Lipschitz constant L, if, x, y in X, (2.12) F (x) F (y) L x y.

3 WEAK SHARP SOLUTIONS OF VARIATIONAL INEQUALITIES The dual gap function for the pseudomonotone VIP. If the mapping F is pseudomonotone, then the solution set of the VIP can be characterized as the intersection of half-spaces, i.e., x is a solution of the VIP if and only if it satisfies (3.1) F (x),x x 0 x X. It follows that the solution set of the VIP is closed and convex. The proof of this result is identical to that given in Auslender [2] for monotone variational inequalities. Note that we cannot substitute quasimonotonicity for pseudomonotonicity in (3.1), as shown by the VIP involving the quasimonotone function F (x) =x 2 and the set X = R. We define the dual gap function G(x) associated with the VIP as (3.2) G(x) = max F (z), x z z X = F (ỹ),x ỹ, where ỹ is any point in the set Λ(x) := arg max z X F (z),x z. Since the function G is the pointwise supremum of affine functions, it is closed and convex on X. Moreover, G is nonnegative and achieves its minimum value (zero) only at points of X that satisfy the original variational inequality. Thus, any solution of the VIP is a global minimum for the convex optimization program (3.3) min G(x). x X If F is pseudomonotone +, the dual gap function G enjoys the nice properties given in the theorem below. Theorem 3.1. Let F be continuous and pseudomonotone + on X. Then (i) F is constant over X ; (ii) for any x in X, F is constant and equal to F (x ) over Λ(x ); (iii) Λ(x )=X for any x in X ; (iv) if X is compact, then G is continuously differentiable over X, and G(x )= F (x ) x in X. Proof. (i) Let x and x be any two solutions of the VIP. It follows from (2.1) and (3.1) that from which we deduce Now, the inequality F (x ),x x 0, F (x ),x x 0, F (x ),x x =0. F (x ),x x 0, together with the pseudomonotonicity + of F, yields F (x )=F (x ). (ii) For every x X and y in Λ(x ), (3.4) G(x )= F (y ),x y =0 holds. Now, F (x ),y x 0(x is a solution of the VIP) and F (y ),y x =0 imply, by the pseudomonotonicity + of F, that F (x )=F (y ).

4 182 PATRICE MARCOTTE AND DAOLI ZHU (iii) Let x Λ(x ). From (ii) we have that F (x )=F ( x) and F ( x),x x =0. This implies that, for any y in X, (3.5) (3.6) (3.7) F ( x), x y = F ( x), x x + F ( x),x y = F (x ),x y 0 and x is in X. Conversely, for any x in X, (3.8) (3.9) x X F ( x),x x =0 x Λ(x ). (iv) From a result of Danskin [4] the derivative of G at x in the direction d is given by the expression G (x ; d) = max{ F (y),d : y Λ(x )} = F (x ),d, since, by (ii), F is constant and equal to F (x ) over Λ(x ). Thus, G is continuously differentiable at every point x X, with gradient G(x )=F (x ). 4. Sharp solutions of variational inequalities. Recently, in the context of convex smooth optimization, Burke and Ferris [5] have extended the notion of a strongly unique solution to optimization problems whose solution set is not necessarily a singleton. To this aim, they introduced the notion of a weak sharp solution for a convex minimization problem. We recall that the solution set X is weakly sharp for the program min x X f(x) if there exists a positive number α (modulus of sharpness) such that (4.1) f(x) f(x )+α dist(x, X ) x X, where dist(x, X ):=min x X x x. These authors proved that if f is a closed, proper, and convex function and if the sets X and X are nonempty, closed, and convex, then the solution set of the convex optimization program (4.1) is weakly sharp if and only if the geometric condition ( ) (4.2) f(x ) int x X x X [T X (x) N X (x)] holds. Since the VIP lacks a natural objective function, it is natural to define weak sharpness of the solution set of a variational inequality with reference to (4.2). Precisely, following Patriksson [14], we say that the solution set of the VIP is weakly sharp if we have, for any x in X, ( ) (4.3) F (x ) int. x X [T X (x) N X (x)] Alternatively, one could have defined weak sharpness with respect to an artificial convex programming reformulation of the VIP. If F is pseudomonotone, an obvious choice for such a reformulation is the one based on the dual gap function defined earlier (see (3.2) and (3.3)). This would have led to the definition

5 WEAK SHARP SOLUTIONS OF VARIATIONAL INEQUALITIES 183 (4.4) G(x) α dist(x, X ) x in X. If this condition is fulfilled, the function G provides an error bound for the distance from a feasible point to the set of solutions to the VIP. The constant α is again called the modulus of sharpness for the solution set X. Note that the very evaluation of G at a point x requires the solution of a possibly nonconvex mathematical program. From this point on, we will adopt the geometric condition (4.3) as the definition of weak sharpness and show that both definitions are actually equivalent whenever F is pseudomonotone +. Theorem 4.1. Let F be continuous and pseudomonotone + over the compact set X. Let the solution set X of the VIP be nonempty. Then X is weakly sharp if and only if there exists a positive number α such that G(x) α dist(x, X ) x in X. Proof. Let B denote the unit ball in R n. We first prove that the inclusion (4.5) αb F (x )+[T X (x ) N X (x )] holds at x X if and only if we have (4.6) F (x ),z α z z T X (x ) N X (x ). Indeed, if (4.5) holds, then for every y B, wehave (4.7) αy F (x ) [T X (x ) N X (x )]. Thus, for every z [T X (x ) N X (x )], we have αy F (x ),z 0. Taking y = z/ z in the above inequality, we obtain (4.6). Now assume that (4.6) holds. Then there exists a positive number α such that, for x X, y B, and z T X (x ) N X (x ), F (x )+αy, z = F (x ),z + α y, z F (x ),z + α y z F (x ),z + α z 0 by (4.6). This implies that (4.5) holds as well. If F (x ) int ( x X [T X(x) N X (x)] ) x in X, then there must exist a positive number α such that (4.5) is satisfied for every x X. From the above derivation, we have that F (x ),z α z for every z in T X (x ) N X (x ). Now set, for x in X, x = proj X (x). Clearly, x x T X ( x) N X ( x) and there follows F ( x),x x α x x = α dist(x, X ). Since G is a convex function, differentiable at x X,wehave G(x) =G(x) G( x) G( x),x x = F ( x),x x α dist(x, X ).

6 184 PATRICE MARCOTTE AND DAOLI ZHU Conversely, let X satisfy (4.4) for some positive number α and let x be a point in X. If T X (x ) N X (x )={0}, then [T X (x ) N X (x )] = R n and αb F (x )+[T X (x ) N X (x )], trivially. Otherwise, let d be a nonzero vector in T X (x ) N X (x ). For any y X,wehave d, y x 0 since d T X (x ), d, y x 0 since d N X (x ). Those inequalities imply that d, y x = 0, and X is a subset of a hyperplane H d orthogonal to d. Let {d k } be a sequence converging to d such that x + t k d k X for some sequence of positive numbers {t k }. We can write dist(x + t k d k,x ) dist(x + t k d k,h d ) = t k d, d k. d Since X satisfies (4.4) with modulus α, we obtain and G(x + t k d k ) α dist(x + t k d k,x ) αt k d, d k d (G(x + t k d k ) G(x ))/t k α d, dk d. Taking the limit as t k 0 and d k d leads to G(x ),d α d d in T X (x ) N X (x ). Therefore, for any w in B, αw F (x ),d = αw, d G(x ),d α d α d =0, and it follows that αb F (x )+[T X (x ) N X (x )]. Since F is constant over X, we conclude that (4.3) holds. We now show, by means of an example, that pseudomonotonicity of F is too weak a condition for the above result to hold. Indeed, consider the variational inequality defined by the two-dimensional mapping F (x) =( x 2, 2x 1 ) and the set X = {0 x 1 1, 0 x 2 1}. One can check that the mapping F is pseudomonotone but not pseudomonotone + on X. Indeed, F is not constant over its solution set X = {x X : x 2 =0}, in contradiction with the first statement of Theorem 3.1. We have G(x) = max y X ( y 2, 2y 1 ), (x 1 y 1,x 2 y 2 ) = max y X x 1y 2 y 1 y 2 +2x 2 y 1 =2x 2 = 2 dist(x, X ),

7 WEAK SHARP SOLUTIONS OF VARIATIONAL INEQUALITIES 185 and X satisfies (4.4) with modulus α = 2. However, for any x in X, we have [T X (x ) N X (x )] = {x 2 0}. Consequently F (x ) does not lie inside [T X (x ) N X (x )] x X for any x in the solution set X, and the solution set X is not weakly sharp. Our second characterization of weak sharpness involves the notion of minimum principle sufficiency introduced by Ferris and Mangasarian [6]. Consider the reformulation of the VIP as the (possibly nonconvex and/or nonsmooth) optimization problem min x X g(x), where the primal gap function g is defined as (4.8) g(x) :=max F(x),x y, y X and let Γ(x) := arg max F (x),x y y X = arg min F (x),y. y X We say that the VIP possesses the minimum principle sufficiency (MPS) property if Γ(x ) coincides with the solution set X, for every x in X. Theorem 4.2. Assume that F is continuous on X and that the set ( ) K := int x X [T X (x) N X (x)] is nonempty. Then, for each z in K, one has that arg max{ z,y : y X} X. Moreover, if F is pseudomonotone and F (x ) K for every x X, then the VIP possesses the MPS property. Proof. Let x X, x/ X, and x = proj X (x). We have that x x T X ( x) N X ( x), and, for any given z in K, there exists a positive number δ such that z + w, x x < 0 w in δb. Thus, z,x < z, x δ x x ; i.e., x/ arg max{ z,y : y X}, which brings about the conclusion by contradiction. Next, let F (x ) K for x X. In the first part of the proof, it has been established that arg max{ F (x ),y : y X} X. Let ˆx be in X. We have, as before, F (x ), ˆx x =0. Now, for any y in X, F (x ), ˆx y = F (x ), ˆx x + F (x ),x y 0. Therefore, ˆx Γ(x ) and X Γ(x ). By gathering the two preceding inclusions, we conclude that arg max{ F (x ),y : y X} = X, as claimed. Theorem 4.3. Let F be pseudomonotone + and continuous on the compact polyhedral set X. Then the VIP possesses the MPS property if and only if it is weakly sharp, i.e., X =Γ(x )=Λ(x ).

8 186 PATRICE MARCOTTE AND DAOLI ZHU Proof. The if part of the statement is a consequence of Theorem 4.2. To prove the converse, first observe that the solution set Γ(x )=X of the linear program min F x X (x ),x is weakly sharp (see appendix in Mangasarian and Meyer [10], for instance) with positive modulus α, and that α only depends on the constant vector F (x ) and X. We develop G(x) = max y X F (y),x y F (x ),x x x X = F (x ),x ˆx ˆx Γ(x ) α x proj Γ(x )(x) = α x proj X (x) α dist(x, X ), and from Theorem 4.1, X is weakly sharp. 5. Finite convergence of algorithms for solving the VIP. In this section we will derive finite convergence results for classes of algorithms under the condition that the solution set of the VIP be weakly sharp. The first such result generalizes a result of Al-Khayyal and Kyparisis [1] to the case where the solution set is not necessarily a singleton. Theorem 5.1. Let F be continuous and pseudomonotone + over the set X, and let the solution set X of the VIP be weakly sharp. Also let {x k } be a sequence in R n. If either (i) the sequence {dist(x k,x )} converges to zero and the mapping F is uniformly continuous on an open set containing the sequence {x k } and the set X,or (ii) the sequence {x k } converges to some x X, then there exists a positive integer k 0 such that, for any index k k 0, any solution of the linear program (5.1) min F x X (xk ),x is a solution of the VIP. Proof. First assume that (i) holds. From Theorem 3.1, F (x ) is constant over X and there must exist a uniform positive constant α such that (5.2) F (x )+αb x X [T X (x) N X (x)] for every x in X. Since F is uniformly continuous and dist(x k,x ) 0, there exists an integer k 0 such that F (x k ) F (x ) <α k k 0, i.e., F (x k ) int ( x X [T X(x) N X (x)] ). Therefore, by Theorem 4.2, arg min x X F (xk ),x X. Under condition (ii) the result (5.2) is still valid for every x X, and we obtain the result as a consequence of the convergence of the sequence { F (x k ) F (x ) } k to zero.

9 WEAK SHARP SOLUTIONS OF VARIATIONAL INEQUALITIES 187 If Ω is a nonempty, closed, and convex subset of X, Burke and Ferris [5] have proved the inclusion (5.3) Ω+ [T X (x ) N Ω (x)] [x + N X (x)]. x Ω x Ω We will now use this result to provide a geometric characterization of sequences that achieve the finite identification of a solution to the VIP. Theorem 5.2. Let F be pseudomonotone + and continuous over the compact set X. Let the solution set X of the VIP be weakly sharp. Let {x k } be a subsequence with elements in X such that the real sequence {dist(x k,x )} converges to zero. If F is uniformly continuous on an open set containing {x k } and X, then there exists a positive integer k 0 such that, for any index k k 0, x k is a solution of the VIP if and only if (5.4) lim proj T k X (x )( F (x k ))=0. k Proof. Ifx k X, then F (x k ) N X (x k ) and (5.4) holds trivially. Otherwise, assume that (5.4) is satisfied. The Moreau decomposition of F (x k ) along T X (x) and its polar cone N X (x) yields F (x k ) = proj TX (x)( F (x k )) + proj NX (x)( F (x k )). By Theorem 3.1, we have that F is constant over X. Thus, for any x X, the assumptions imply and so F (x ) + proj NX (x)( F (x k )) 0, dist(x k + proj NX (x k )( F (x k )),X F (x )) 0. But, from the weak sharpness property, one has ( X F (x ) int X + ) [T X (x) N X (x)]. x X Now, for x k close to x in X, we have, using (5.3), ( x k + proj NX (x )( F (x k )) int X + ) [T k X (x) N X (x)] x X [x + N X (x)]. x X Therefore, k sufficiently large, x k = proj X (x k + proj NX (x )( F (x k ))) ( k ) proj X [x + N X (x)] x X x X {x} = X.

10 188 PATRICE MARCOTTE AND DAOLI ZHU This completes the proof. Several authors have proposed general iterative frameworks for solving variational inequalities. For instance, Cohen [3], or Zhu and Marcotte [18] investigated a scheme in which x k+1 is a solution of the variational inequality (5.5) σf(x k )+H(x k+1 ) H(x k ),x x k+1 0 x X, where σ is a positive constant and H is an auxiliary mapping, usually taken to be strongly monotone. Under suitable assumptions on F (strong monotonicity or cocoercivity 1 ) and σ, the sequence {x k } is known to converge to a solution of the original variational inequality. From now on, we restrict our attention to those cases where the algorithm returns a convergent sequence {x k } whose limiting point is a solution of the VIP, and provide a sufficient condition for its finite termination. Lemma 5.1. Let F and H be uniformly continuous on X and {x k } x. Then the sequence {proj TX (x k+1 )( F (x k+1 ))} converges to zero. Proof. Since the sequence {x k } is convergent, x k+1 x k 0. From (5.5), we have [σf(x k )+H(x k+1 ) H(x k )] N X (x k+1 ). The Moreau decomposition technique yields proj TX (x )(F (x k+1 )) = k+1 min v N X (x k+1 ) (x k+1 ) v = min z z F (x k+1 ) N X (x k+1 ) [F (xk+1 ) F (x k )] + 1 σ [H(xk+1 ) H(x k )]. From the uniform continuity of F and H, the right-hand side of the above inequality converges to zero, and we obtain that {proj TX (x )( F (x k+1 )} converges to zero, as k+1 claimed. Combining Theorem 5.2 and Lemma 5.1, we obtain the following result. Theorem 5.3. Under the assumptions of Theorem 5.2 and Lemma 5.1, the general iterative algorithm for solving the VIP based on the auxiliary problem (5.5) generates a sequence {x k } such that, for all k sufficiently large, x k is a solution of the VIP. Recently, Zhu and Marcotte [18] have proposed a descent framework for the VIP, based on the auxiliary variational inequality (5.5), that includes as particular cases Fukushima s projective method [7] and Taji, Fukushima, and Ibaraki s Newton method [16] (see also Fukushima [8] or Larsson and Patriksson [9] for a survey of descent methods for the VIP). Given a mapping H(w, x) defined on X X, continuous and strongly monotone with respect to the variable x and such that H(x, x) coincides with the values F (x) of the original mapping, a direction d k is specified, at iteration k, as d k = w k x k, where w k is the unique solution of the auxiliary variational inequality (5.6) H(w k,x k ) H(x k,x k ),x x k 0 x X. 1 The mapping F is co-coercive on the set X if there exists a positive number β such that F (x) F (y),x y β F (x) F (y) 2 for all x, y in X; i.e., its inverse mapping is strongly monotone.

11 WEAK SHARP SOLUTIONS OF VARIATIONAL INEQUALITIES 189 The iterate x k+1 is then obtained by minimizing some merit function (related to the auxiliary mapping H) along the direction d k. Under a suitable assumption, we have that d k is a descent direction for the merit function at the point x k, and it can be shown that d k converges to zero, while both w k and x k converge to a solution of the VIP. If F and H are both uniformly continuous, respectively, on X and X X, we then obtain that proj TX (w k )( F (w k )) converges to zero. If, furthermore, the assumptions of Theorem 5.2 are satisfied, the sequence {x k } converges to x after a finite number of iterations. REFERENCES [1] F. A. Al-Khayyal and J. Kyparisis, Finite convergence of algorithms for nonlinear programming and variational inequalities, J. Optim. Theory Appl., 70 (1991), pp [2] A. Auslender, Optimisation, Méthodes numériques, Masson, Paris, [3] G. Cohen, Auxiliary problem principle extended to variational inequalities, J. Optim. Theory Appl., 59 (1988), pp [4] J. M. Danskin, The theory of min-max with applications, SIAM J. Appl. Math., 14 (1966), pp [5] J. V. Burke and M. C. Ferris, Weak sharp minima in mathematical programming, SIAM J. Control Optim., 31 (1993), pp [6] M. C. Ferris and O. L. Mangasarian, Minimum principle sufficiency, Math. Programming, 57 (1992), pp [7] M. Fukushima, Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems, Math. Programming, 53 (1992), pp [8] M. Fukushima, Merit functions for variational inequality and complementarity problems, Nonlinear Optimization and Applications, G. Di Pillo and F. Giannessi, eds., Plenum Press, 1996, pp [9] T. Larsson and M. Patriksson, A class of gap functions for variational inequalities, Math. Programming, 64 (1994), pp [10] O. L. Mangasarian and R. R. Meyer, Nonlinear perturbation of linear programming, SIAM J. Control Optim., 17 (1979), pp [11] P. Marcotte and D. L. Zhu, Monotone + mappings and variational inequalities, in Fifth International Symposium on Generalized Convexity, Luminy, France, June [12] J. J. Moreau, Décomposition orthogonale dans un espace hilbertien selon deux cônes mutuellement polaires, Comptes Rendus Hebdomadaires des Séances de l Académie des Sciences (Paris), Série A, 255 (1962), pp [13] S. Nguyen and C. Dupuis, An efficient method for computing traffic equilibria in networks with asymmetric transportation costs, Transportation Sci., 18 (1984), pp [14] M. Patriksson, A unified framework of descent algorithms for nonlinear programs and variational inequalities, Ph.D. thesis, Department of Mathematics, Linköping Institute of Technology, Linköping, Sweden, [15] S. Schaible, Generalized monotonicity, in Proceedings of the 10th International Summer School on Nonsmooth Optimization, Analysis and Applications, Erice, Italy, 1991, F. Giannessi, ed., Gordon and Breach, Amsterdam, The Netherlands, [16] K. Taji, M. Fukushima, and T. Ibaraki, A globally convergent Newton method for solving strongly monotone variational inequalities, Math. Programming, 58 (1993), pp [17] D. L. Zhu and P. Marcotte, New classes of generalized monotononicity, J. Optim. Theory. Appl., 87 (1995), pp [18] D. L. Zhu and P. Marcotte, An extended descent framework for variational inequalities, J. Optim. Theory. Appl., 80 (1994), pp [19] D. L. Zhu and P. Marcotte, Co-coercivity and its role in the convergence of iterative schemes for solving variational inequalities, SIAM J. Optim., 6 (1996), pp