A proximal algorithm for constrained multiobjective minimization with quasiconvex functions

Abstract

In this paper, an inexact proximal point algorithm with proximal distances is introduced to solve multiobjective minimization problems with locally Lipschitz quasiconvex objective functions constrained to a closed convex set with nonempty interior. We prove the convergence of the sequence generated by the algorithm to a Pareto-Clarke critical point and convergence to a Pareto solution when the objective functions are convex or when we consider the exact proximal algorithm for the quasiconvex problem when the proximal parameter converges to zero. We give some conditions to obtain the finite termination of the algorithm and prove linear or superlinear rate of convergence for a large class of proximal distances. Thus, this work extends the convergence of the proximal point method for quasiconvex multiobjective minimization problems and improves recent convergence results of previous works for the convex case. Finally, we give numerical experiments to confer the practicability of the algorithm.