A Linear Scalarization Proximal Point Method for Quasiconvex Multiobjective Minimization

TL;DR

This paper introduces a linear scalarization proximal point method for quasiconvex multiobjective minimization, proving convergence to critical points and Pareto solutions under various conditions.

Contribution

It develops a novel algorithm with convergence guarantees for quasiconvex multiobjective problems, including inexact and special convex cases.

Findings

Convergence to generalized critical points under natural assumptions.

Convergence to weak Pareto solutions when proximal parameters tend to zero.

Finite convergence to Pareto optimal points in convex cases with sharp minima.

Abstract

In this paper we propose a linear scalarization proximal point algorithm for solving arbitrary lower semicontinuous quasiconvex multiobjective minimization problems. Under some natural assumptions and using the condition that the proximal parameters are bounded we prove the convergence of the sequence generated by the algorithm and when the objective functions are continuous, we prove the convergence to a generalized critical point. Furthermore, if each iteration minimize the proximal regularized function and the proximal parameters converges to zero we prove the convergence to a weak Pareto solution. In the continuously differentiable case, it is proved the global convergence of the sequence to a Pareto critical point and we introduce an inexact algorithm with the same convergence properties. We also analyze particular cases of the algorithm obtained finite convergence to a Pareto optimal point when the objective functions are convex and a sharp minimum condition is satisfied.