The inexact projected gradient method for quasiconvex vector optimization problems

TL;DR

This paper introduces an inexact projected gradient method for solving smooth constrained vector optimization problems with a focus on $K$-quasiconvex objectives, proving its global convergence to stationary points.

Contribution

It extends existing methods by proving convergence under $K$-quasiconvexity, a weaker condition than $K$-convexity, for vector optimization problems.

Findings

Proves global convergence of the method to stationary points.

Applicable to $K$-quasiconvex functions, broadening the scope.

Provides a practical approach for real-world vector optimization problems.

Abstract

Vector optimization problems are a generalization of multiobjective optimization in which the preference order is related to an arbitrary closed and convex cone, rather than the nonnegative octant. Due to its real life applications, it is important to have practical solution approaches for computing. In this work, we consider the inexact projected gradient-like method for solving smooth constrained vector optimization problems. Basically, we prove global convergence of any sequence produced by the method to a stationary point assuming that the objective function of the problem is $K$-quasiconvex, instead of the stronger $K$-convexity assumed in the literature.