Stopping Rules for Gradient Method for Saddle Point Problems with Twoside Polyak-Lojasievich Condition

TL;DR

This paper introduces a stopping rule for gradient methods solving saddle point problems under a two-sided Polyak-Lojasievich condition, ensuring solution quality with inexact gradients.

Contribution

It proposes a novel stopping rule based on the inexact gradient norm and demonstrates its effectiveness through numerical experiments.

Findings

Effective stopping rule ensures solution quality

Numerical experiments confirm convergence and efficiency

Comparison with existing methods shows improved performance

Abstract

The paper considers approaches to saddle point problems with a two-sided variant of the Polyak-Lojasievich condition based on the gradient method with inexact information and proposes a stopping rule based on the smallness of the norm of the inexact gradient of the external subproblem. Achieving this rule in combination with a suitable accuracy of solving the auxiliary subproblem ensures that the quality of the original saddle point problem is acceptable. The results of numerical experiments for various saddle point problems are discussed to illustrate the effectiveness of the proposed method, including the comparison with proven convergence rate estimates.