A Steepest Gradient Method with Nonmonotone Adaptive Step-sizes for the Nonconvex Minimax and Multi-Objective Optimization Problems

TL;DR

This paper introduces a new gradient descent method with adaptive step sizes for nonconvex minimax and multi-objective problems, proving convergence and demonstrating effectiveness through numerical experiments.

Contribution

It presents a novel steepest gradient method with nonmonotone adaptive step sizes for nonconvex minimax and multiobjective optimization, with proven convergence results.

Findings

Convergence to weakly efficient or Pareto stationary solutions is established.

Numerical experiments verify theoretical convergence and effectiveness.

The method effectively handles nonconvex, quasiconvex, and pseudoconvex functions.

Abstract

This paper proposes a new steepest gradient descent method for solving nonconvex finite minimax problems using non-monotone adaptive step sizes and providing proof of convergence results in cases of the nonconvex, quasiconvex, and pseudoconvex differentiate component functions. The proposed method is applied using a referenced-based approach to solve the nonconvex multiobjective programming problems. The convergence to weakly efficient or Pareto stationary solutions is proved for pseudoconvex or quasiconvex multiobjective optimization problems, respectively. A variety of numerical experiments are provided for each scenario to verify the correctness of the theoretical results corresponding to the algorithms proposed for the minimax and multiobjective optimization problems.