The proximal point method for locally lipschitz functions in multiobjective optimization with application to the compromise problem

This paper studies the constrained multiobjective optimization problem of finding Pareto critical points of vector-valued functions. The proximal point method considered by Bonnel, Iusem, and Svaiter [SIAM J. Optim., 15 (2005), pp. 953–970] is extended to locally Lipschitz functions in the finite dimensional multiobjective setting. To this end, a new (scalarization-free) approach for convergence analysis of the method is proposed where the first-order optimality condition of the scalarized problem is replaced by a necessary condition for weak Pareto points of a multiobjective problem. As a consequence, this has allowed us to consider the method without any assumption of convexity over the constraint sets that determine the vectorial improvement steps. This is very important for applications; for example, to extend to a dynamic setting the famous compromise problem in management sciences and game theory.Fundação de Amparo à Pesquisa do Estado de GoiásConselho Nacional de Desenvolvimento Científico e TecnológicoCoordenação de Aperfeiçoamento de Pessoal de Nivel SuperiorMinisterio de Economía y CompetitividadAgence nationale de la recherch

Hybrid Approximate Proximal Method with Auxiliary Variational Inequality for Vector Optimization

This paper studies the general vector optimization problem of finding weakly efficient points for mappings in a Banach space Y, with respect to the partial order induced by a closed, convex, and pointed cone C C Y with nonempty interior. In order to find a solution of this problem, we introduce an auxiliary variational inequality problem for monotone, Lipschitz-continuous mapping. The approximate proximal method in vector optimization is extended to develop a hybrid approximate proximal method for the general vector optimization problem by the combination of extragradient method for finding a solution to the variational inequality problem and approximate proximal point method for finding a root of a maximal monotone operator. In this hybrid approximate proximal method, the subproblems consist of finding approximate solutions to the variational inequality problem for monotone, Lipschitz-continuous mapping, and finding weakly efficient points for suitable regularizations of the original mapping. We present both an absolute and a relative version in which the subproblems are solved only approximately. Weak convergence of the generated sequence to a weak efficient point is established under quite mild conditions. In addition, we also discuss an extension to Bregman-function-based hybrid approximate proximal algorithms for finding weakly efficient points for mappings