A proximal subgradient algorithm for constrained multiobjective DC-type optimization

TL;DR

This paper introduces a proximal subgradient algorithm designed for constrained multiobjective optimization problems involving difference-of-convex functions, providing theoretical optimality conditions and convergence guarantees.

Contribution

It develops a novel algorithm for multiobjective DC problems and establishes necessary and sufficient optimality conditions for such problems.

Findings

Algorithm generates bounded sequences with cluster points as stationary solutions

Provides theoretical foundation for multiobjective DC optimization

Convergence under mild assumptions

Abstract

In this paper, we consider a class of constrained multiobjective optimization problems, where each objective function can be expressed by adding a possibly nonsmooth nonconvex function and a differentiable function with Lipschitz continuous gradient, then subtracting a weakly convex function. This encompasses multiobjective optimization problems involving difference-of-convex (DC) functions, which are prevalent in various applications due to their ability to model nonconvex problems. We first establish necessary and sufficient optimality conditions for these problems, providing a theoretical foundation for algorithm development. Building on these conditions, we propose a proximal subgradient algorithm tailored to the structure of the objectives. Under mild assumptions, the sequence generated by the proposed algorithm is bounded and each of its cluster points is a stationary solution.