The Proximal Point Method for Locally Lipschitz Functions in Multiobjective Optimization with Application to the Compromise Problem

Abstract

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.

A proximal point method for difference of convex functions in multi-objective optimization with application to group dynamic problems

“…Our non-convex descent property also implies the one obtained by Ji et al [34]. It is worth to mention that even if DC function is a locally Lipschitz function our method is different to the one considered by Bento et al [4] because our algorithm minimizes at each iteration a linear approximation of the objective function instead of minimizing the objetive function as [4] does. As we will see in Section 6 our approach have important applications in Behavioral Sciences.…”

“…Remark 8 It is known that every DC function is a locally Lipschitz function; see [30]. The proximal point method for locally Lipschitz functions in multiobjective optimization was analyzed by Bento et al [4] who firstly considered a possibly non-convex vector improving constraints Ω k . However, we mention that Algorithm 1 (applied for DC functions) offers an additional flexibility for considering, at each step, a linear approximation G(x) − JH(x k )(x − x k ) instead of minimize the non-convex function…”

“…It also has been shown how our proximal method can greatly help to solve a "group dynamic" problem by using the recent variational rationality approach. In view of Remarks 4 and 6, it seems to be interesting to extend the approach proposed in [4] to infinite dimensional space and consider the concepts of relative minimizers in multi-objective optimization given by Mordukhovich [47,48] which generalize the concepts of Pareto and weak Pareto points. We will consider this topic in a future research.…”

“…Convergence properties of a proximal point method for vector-valued DC functions adding a (non-convex) vectorial improving constraint have been proved. Bento et al [4] handled with a proximal algorithm for multi-objective optimization involving locally Lipchitz vector functions and (non-convex) vectorial improving constraints. Although we use the same approach of convergence analysis our method is different to the one considered in [4] because it minimizes at each iteration a convex approximation instead of the (non-convex) objective function.…”

Iteration-Complexity and Asymptotic Analysis of Steepest Descent Method for Multiobjective Optimization on Riemannian Manifolds

“…. , f m , that have to be optimized at the same time on M. In recent years, there has been a significant increase in the number of papers addressing this class of problems; for example, see [1][2][3][4][5][6][7]. Here, among the methods designed for solving multiobjective optimization problems, we are interested in the steepest descent method.…”