JOURNAL OF INDUSTRIAL AND Website: http://AIMsciences.org MANAGEMENT OPTIMIZATION Volume 1, Number 2, May 2005 pp. 219{233 A DISCRETIZATION BASED SMOOTHING METHOD FOR SOLVING SEMI-INFINITE VARIATIONAL INEQUALITIES Burcu Ozc»amÄ Department of Industrial Engineering North Carolina State University Raleigh, NC 27695-7906, USA Hao Cheng SAS Institute Inc. Cary, NC 27513, USA Abstract. We propose a new smoothing technique based on discretization to solve semi-in¯nite variational inequalities. The proposed algorithm is tested by both linear and nonlinear problems and proven to be e±cient. 1. Introduction. Let Rn be the n-dimensional Euclidean space. Given a function F from Rn into itself and a nonempty subset X of Rn, the ¯nite dimensional variational inequality problem VI(X; F ) is de¯ned as follows. VI(X ; F ) : Find a vector x¤ 2 X such that F (x¤)T (x ¡ x¤) ¸ 0 for all x 2 X: After originating in [9], the theory, algorithms and applications of ¯nite dimensional variational inequalities have been well studied for over four decades. Interested readers can refer to [5] for a comprehensive study. However, most of the algorithms for solving VI(X; F ) in the literature work only for special cases with X possessing certain geometric structure (such as compact polyhedral set) or de¯ned by a ¯nite number of equality or inequality constraints. In this paper, we focus on the semi-in¯nite variational inequality problem SIVI(X; F ) as de¯ned by Fang et al. [8], in which the set X is de¯ned by X = fx 2 Rnjg(x; t) ¸ 0 for all t 2 T g; (1) where T is a nonempty compact subset of Rl. Note that T may contain in¯nitely many elements. For any t 2 T , we assume that g(x; t): Rn £ Rl ! Rp is a continu- ously di®erentiable and concave function. Therefore, X is a convex set in Rn. We further assume that X is nonempty and bounded, which implies compactness. The solvability of ¯nite dimensional variational inequalities and consequently of semi-in¯nite variational inequalities is guaranteed by the following existence results in Hartman and Stampacchia [9]. 2000 Mathematics Subject Classi¯cation. 90C34, 57R12, 49M25. Key words and phrases. Variational inequalities, semi-in¯nite, smoothing. 219 220 BURCU OZC»AMÄ AND HAO CHENG Theorem 1.1. Let X be a compact convex set in Rn and F (x) a continuous map of X into Rn. Then there exists a solution to the variational inequalities VI(X; F ). Proof. Proof can be found in Hartman and Stampacchia [9]. When the set X is de¯ned by in¯nitely many equalities and inequalities, ¯nding the solution of ¯nite dimensional variational inequalities becomes more challeng- ing because general convex programming methods cannot consider in¯nitely many constraints all at the same time. The work by Fang et al. [8] proposes an analytic center cutting plane method to solve SIVI(X; F ). An "-optimal solution is obtained under certain conditions. In a more recent work, Fang et al. [7] consider linear semi- in¯nite variational inequality problem and propose a discretization method followed by an analytic center based inexact cutting plane method. In both studies, pro- posed algorithms are based on nonempty relative interior assumption on X. In this paper, we consider SIVI(X; F ) without nonempty relative interior as- sumption on X. We propose to solve a sequence of variational inequality problem VI(Xi;F ) with Xi de¯ned by a ¯nite number of constraints to approximate the so- lution of SIVI(X; F ). For each VI(Xi;F ), its equivalent KKT formulation, which is nondi®erentiable, is smoothed and solved by Newton's method. A new smoothing function to approximate maxf0; xg is designed to serve this purpose. This paper is organized as follows. The discretization approach is discussed in Section 2. Section 3 presents the smoothing method, which is followed by algo- rithm and the convergence proof in Section 4. Finally, the numerical examples and computational results are presented in Section 5. 2. Discretization. In order to solve the SIVI(X; F ), we use discretization ap- proach to approximate the feasible set X = fx 2 Rnjg(x; t) ¸ 0 for all t 2 T g of our semi-in¯nite variational inequality problem. The cardinality of set T is denoted as jT j and we assume jT j = 1. We can construct a nested sequence fTig of ¯nite subsets of T with the property l that for each i, Ti ½ Ti+1. Since T is a compact subset of R , Ti can satisfy the following assumption. Assumption 2.1. Let ¢ : N ! R be a positive-valued, strictly monotone decreasing function such that limn!1 ¢(n) = 0. Then for each n 2 N and each t 2 T , there 0 exists n0 2 N and t 2 Tn0 such that kt ¡ t0k · ¢(n): De¯ning the ¯nite set Xi as n Xi = fx 2 R jg(x; t) ¸ 0 for all t 2 Tig; (2) We further make the assumption that X1 is bounded. Then each Xi is a compact set. Since X ½ Xi for any i 2 N, from nonempty assumption of X, we know that Xi is a nonempty set consisting of jTij = N constraints. DISCRETIZATION BASED SMOOTHING METHOD 221 We consider the following variational inequality problem VI(Xi;F ) de¯ned set Xi i VI(Xi;F ) : ¯nd x 2 Xi such that i T i F (x ) (x ¡ x ) ¸ 0 for all x 2 Xi: (3) Since Ti ½ Ti+1, we have X ½ Xi+1 ½ Xi. Given that F is continuous and Xi is compact and convex, the solution to (3) is guaranteed to exist. i Let x be a solution of VI(Xi;F ). The following result guarantees the existence of a subsequence of fxig converging to the solution of SIVI(X; F ). Theorem 2.1. The sequence fxig has at least one accumulation point that solves SIVI(X; F ). i ki Proof. Since fx g ½ X1, a compact set, there exists a subsequence fx g such that lim fxki g =x: ^ i!1 We prove in the following thatx ^ solves SIVI(X; F ), i.e., F (^x)T (x ¡ x^) ¸ 0 for all x 2 X: In fact, ifx ^ is not a solution of SIVI(X; F ), then there exists at least onex ¹ such that F (^x)T (¹x ¡ x^) < 0: ¹ Since F is continuous, there exists k 2 fkig such that ¹ ¹ F (xk)T (¹x ¡ xk) < 0: k¹ However, sincex ¹ 2 X ½ Xk¹ and x is a solution of VI(Xk¹;F ), we have ¹ ¹ F (xk)T (¹x ¡ xk) ¸ 0; which is a contradiction. Thus, for all x 2 X; F (^x)T (x ¡ x^) ¸ 0. The two basic approaches for solving ¯nite dimensional variational inequality problems are solving the KKT conditions of the VI(X; F ) and direct methods. The basic idea of the direct methods is to ¯nd a sequence fxkg in X such that each xk+1 solves VI(X; F k), where F k(x) is some approximation to F (x). Based on this approximation, Newton-based iterative algorithms can be found in [12]. Another direct method is the use of merit functions. Although the use of merit functions is theoretically sound, the severe drawbacks arise in the evaluation of merit functions [5]. As long as the function F is de¯ned everywhere, methods based on the KKT conditions o®er a convenient approach mainly because KKT conditions can easily be reformulated as a mixed complementarity problem. This allows the implementation of equation-based, interior point and smoothing-based solution algorithms, which usually possess sharp convergence results. Among the several algorithms to solve KKT equations are Quasi-Newton method by Qi and Jiang [14], B-di®erentiable Newton method by Pang [13] and trust region method by Yang et al. [17]. The smoothing methods and their fast convergence results can be found in Chen, Qi and Sun's work [2] and [5]. In this paper, our solution approach to VI(X; F ) is based on KKT conditions with min function. By smoothing the min function, we reformulate VI(X; F ) as a smooth system of equations. Thus, the reformulated smoothed problem possesses good potential to avoid the failure of constraint quali¯cations, allowing the direct 222 BURCU OZC»AMÄ AND HAO CHENG application of existing nonlinear programming algorithms. The smoothing approach has been proven to have superior performance for solving mixed complementarity problems by the numerical results of Billups, Dirske and Ferris [1]. 3. Smoothing Approach. In this section, we ¯rst introduce an equivalent mixed nonlinear complementarity formulation of approximation problems VI(Xi;F ). Then present a procedure for implementing a new smoothing approach. Following theo- rem and its proof represents the KKT system of the VI(Xi;F ) as in [5]. Theorem 3.1. If g is continuously di®erentiable and it satis¯es the linear indepen- dence constraint quali¯cation, then solving the approximation problem VI(Xi;F ) is equivalent to solving its Lagrangian problem by the following two statements. ¤ (i) If x solves VI(Xi;F ) and constraint quali¯cation holds for set Xi at the point ¤ ¤ m x , then there exists a vector ¸ 2 R ; m = pjTij; such that ¤ Pm ¤ ¤ F (x ) ¡ j=1 ¸j rgj(x ; t) = 0; ¤ ¤ ¸j gj(x ; t) = 0 j = 1; : : : ; m; ¤ ¤ (4) ¸j ¸ 0; gj(x ; t) ¸ 0 j = 1; : : : ; m: (ii) Conversely, if g(x; t) is concave and if (x¤; ¸¤) satis¯es (4) then x¤ solves the VI(Xi;F ). Proof. See [5], p. 19. Thus, under our assumptions and by Theorem 3.1, KKT conditions in (4) and the problem VI(Xi;F ) are equivalent.