Multiobjective optimization problems, vector variational inequalities and proximal-type methods

Abstract

The main purpose of this thesis is to study the asymptotical properties of multiobjective optimization (also known as vector optimization) and vector variational inequalities. Based on these asymptotical properties, we construct some proximal-type methods for solving convex multiobjective optimization problems and weak vector variational inequality problems. We consider a convex vector optimization problem of finding weak Pareto optimal solutions for an extended vector-valued map from a uniformly convex and uniformly smooth Banach space to a real Banach space, with the latter being ordered by a closed, convex and pointed cone with nonempty interior. For this problem, we develop an extension of the classical proximal point method for the scalar-valued convex optimization. In this extension, the subproblems involve the finding of weak Pareto optimal solutions for some suitable regularizations of the original map by virtue of a Lyapunov functional. We present both exact and inexact versions. In the latter case, the subproblems are solved only approximately within an exogenous relative tolerance. In both cases, we prove weak convergence of the sequences generated by the subproblems to a weak Pareto optimal solution of the vector optimization problem. We also construct a generalized proximal point algorithm to find a weak Pareto optimal solution of minimizing an extended vector-valued map with respect to the positive orthant in finite dimensional spaces. In this extension, the subproblems involve finding weak Pareto optimal solutions for the regularized map by employing a vector-valued Bregman distance function. We prove that the sequence generated by this method converges to a weak Pareto optimal solution of the multiobjective optimization problem by assuming that the original multiobjective optimization problem has a nonempty and compact weak Pareto optimal solution set. We formulate a matrix-valued proximal-type method to solve a weak vector variational inequality problem with respect to the positive orthant in finite dimensional spaces through normal mappings. We also carry out convergence analysis on the method and prove the convergence of the sequences generated by the matrix-valued proximal-type method to a solution of the original problem under some mild conditions. Finally, we investigate the nonemptiness and compactness of the weak Pareto optimal solution set of a multiobjective optimization problem with functional constraints via asymptotic analysis. We then employ the obtained results to derive the necessary and sufficient conditions of the weak Pareto optimal solution set of a parametric multiobjective optimization problem. The study of this thesis has used tools from nonlinear functional analysis, multiobjective programming theory, vector variational inequality theory, asymptotical analysis and numerical linear algebra.