Abstract. We study the convergence properties of a (block) coordinate descent method applied to minimize a nondifferentiable (nonconvex) function f(x1,...,xN) with certain separability and regularity properties. Assuming that f is continuous on a compact level set, the subsequence convergence of the iterates to a stationary point is shown when either f is pseudoconvex in every pair of coordinate blocks from among NA1 coordinate blocks or f has at most one minimum in each of NA2 coordinate blocks. If f is quasiconvex and hemivariate in every coordinate block, then the assumptions of continuity of f and compactness of the level set may be relaxed further. These results are applied to derive new (and old) convergence results for the proximal minimization algorithm, an algorithm of Arimoto and Blahut, and an algorithm of Han. They are applied also to a problem of blind source separation. Key Words. Block coordinate descent, nondifferentiable minimization, stationary point, Gauss–Seidel method, convergence, quasiconvex functions,

We present a survey of nondifferentiable optimization problems and methods with special focus on the analytic center cutting plane method. We propose a self-contained convergence analysis, that uses the formalism of the theory of self-concordant functions, but for the main results, we give direct proofs based on the properties of the logarithmic function. We also provide an in depth analysis of two extensions that are very relevant to practical problems: the case of multiple cuts and the case of deep cuts. We further examine extensions to problems including feasible sets partially described by an explicit barrier function, and to the case of nonlinear cuts. Finally, we review several implementation issues and discuss some applications.

. Recently, Peng considered a merit function for the variational inequality problem (VIP), which constitutes an unconstrained dierentiable optimization reformulation of VIP. In this paper, we generalize the merit function proposed by Peng and study various properties of the generalized function. We call this function the D-gap function. We give conditions under which any stationary point of the D-gap function is a solution of VIP and conditions under which it provides a global error bound for VIP. We also present a descent method for solving VIP based on the D-gap function. Key words: Variational inequality problems, unconstrained optimization reformulation, global error bound, descent method. 1 The authors are grateful to C. Kanzow for his comments on an earlier version of the paper. They also thank the referees for their constructive comments. y The work of this author was supported by the Research Fellowships of the Japan Society for the Promotion of Science for Young Scientist...

. We propose new methods for solving the variational inequality problem where the underlying function F is monotone. These methods may be viewed as projection-type methods in which the projection direction is modified by a strongly monotone mapping of the form I \Gamma ffF or, if F is affine with underlying matrix M , of the form I + ffM T , with ff 2 (0; 1). We show that these methods are globally convergent and, if in addition a certain error bound based on the natural residual holds locally, the convergence is linear. Computational experience with the new methods is also reported. Key words. Monotone variational inequalities, projection-type methods, error bound, linear convergence. AMS subject classifications. 49M45, 90C25, 90C33 1. Introduction. We consider the monotone variational inequality problem of finding an x 2 X satisfying F (x ) T (x \Gamma x ) 0 8x 2 X; (1) where X is a closed convex set in ! n and F is a monotone and continuous function from ! n to ...

Abstract. We propose a new projection algorithm for solving the variational inequality problem, where the underlying function is continuous and satisfies a certain generalized monotonicity assumption (e.g., it can be pseudomonotone). The method is simple and admits a nice geometric interpretation. It consists of two steps. First, we construct an appropriate hyperplane which strictly separates the current iterate from the solutions of the problem. This procedure requires a single projection onto the feasible set and employs an Armijo-type linesearch along a feasible direction. Then the next iterate is obtained as the projection of the current iterate onto the intersection of the feasible set with the halfspace containing the solution set. Thus, in contrast with most other projection-type methods, only two projection operations per iteration are needed. The method is shown to be globally convergent to a solution of the variational inequality problem under minimal assumptions. Preliminary computational experience is also reported. Key words. variational inequalities, projection methods, pseudomonotone maps

. It is well-known (see Pang and Chan [7]) that Newton's method, applied to strongly monotone variational inequalities, is locally and quadratically convergent. In this paper we show that Newton's method yields a descent direction for a nonconvex, nondifferentiable merit function, even in the abscence of strong monotonicity. This result is then used to modify Newton's method into a globally convergent algorithm by introducing a linesearch strategy. Furthermore, under strong monotonicity (i) the optimal face is attained after a finite number of iterations (ii) the stepsize is eventually fixed to the value one, resulting in the usual Newton step. Computational results are presented. Keywords. Mathematical Programming. Variational Inequalities. Newton's method. Research supported by NSERC grants A5789 and A5491. 1. Problem formulation and basic definitions. Let \Phi be a nonempty, convex and compact subset of R n . Consider the variational inequality problem consisting in finding x ...

Recently, Tseng extended several merit functions for the nonlinear complementarity problem to the semidefinite complementarity problem (SDCP) and investigated various properties of those functions. In this paper, we propose a new merit function for the SDCP based on the squared Fischer-Burmeister function and show that it has some favorable properties. Particularly, we give conditions under which the function provides a global error bound for the SDCP and conditions under which it has bounded level sets. We also present a derivative-free method for solving the SDCP and prove its global convergence under suitable assumptions.

This survey gives an introduction to some of the recent developments in the field of complementarity and related problems. After presenting two typical examples and the basic existence and uniqueness results, we focus on some new trends for solving nonlinear complementarity problems. Extensions to mixed complementarity problems, variational inequalities and mathematical programs with equilibrium constraints are also discussed.

We present a framework for descent algorithms that solve the monotone variational inequality problem V IP v which consists in finding a solution v 2\Omega v which satisfies s(v) T (u \Gamma v) 0, for all u 2\Omega v . This unified framework includes, as special cases, some well known iterative methods and equivalent optimization formulations. A descent method is developed for an equivalent general optimization formulation and a proof of its convergence is given. Based on this unified algorithmic framework, we show that a variant of the descent method where each subproblem is only solved approximately is globally convergent under certain conditions. Key words Variational Inequalities, Descent methods, Optimization. 1 Introduction In this paper we consider the variational inequality problem (VIP) that consists in finding a vector v in\Omega v such that (V IP v ) s(v) T (u \Gamma v) 0 for all u 2\Omega v (1) where u, v are vectors in R n and s a mapping from the clos...

. The variational inequality problem is reduced to an optimization problem with a differentiable objective function and simple bounds. Theoretical results are proved, that relate stationary points of the minimization problem to solutions of the variational inequality problem. Perturbations of the original problem are studied and an algorithm that uses the smooth minimization approach for solving monotone problems is defined. Key words. Variational inequalities, box constrained optimization, complementarity. 1 Introduction Let\Omega be a nonempty, closed and convex subset of IR n and F : IR n ! IR n . The finite-dimensional variational inequality problem, denoted by VIP, is to find a vector x 2\Omega such that hF (x); w \Gamma xi 0; for all w 2\Omega : (1) This problem has many interesting applications and its solution using special techniques has been considered extensively in the literature; see, for example, (Ref. 1) and references therein. The linear and nonlinear comp...