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In this article, we consider the convex min-max problem with infinite constraints. We propose an exchange method to solve the problem by using efficient inactive constraint dropping rules. There is no need to solve the maximization problem over the metric space, as the algorithm has merely to find some points in the metric space such that a certain criterion is satisfied at each iteration. Under some mild assumptions, the proposed algorithm is shown to terminate in a finite number of iterations and to provide an approximate solution to the original problem. Preliminary numerical results with the algorithm are promising. To our knowledge, this article is the first one conceived to apply explicit exchange methods for solving nonlinear semi-infinite convex min-max problems. [ABSTRACT FROM AUTHOR]

In this article, we establish some relationships between a solution of generalized vector variational-like inequalities and an efficient solution or a weakly efficient solution to the nonsmooth vector optimization problem under the assumptions of pseudoinvexity or invariant pseudomonotonicity. Our results extend and improve the corresponding results in the literature. [ABSTRACT FROM PUBLISHER]

We develop an iterative method for solving the KKT system of the semi-infinite programming (SIP) problem. At each iteration, we solve the KKT system of a nonlinear programming problem with finite constraints by a semismooth Newton method. The algorithm either terminates at a KKT point of the SIP problem in finitely many iterations or generates an infinite sequence of iterates whose any accumulation point is a KKT point of the problem. We also analyse the convergence rate of the method. Preliminary numerical results are reported. [ABSTRACT FROM AUTHOR]