On the Convergence of Constrained Gradient Method

TL;DR

This paper improves convergence guarantees for the constrained gradient method in convex optimization and variational inequality problems under weaker assumptions, supported by numerical experiments.

Contribution

It provides rigorous convergence analysis of CGM under less restrictive conditions for strongly convex and strongly monotone problems.

Findings

CGM converges under weaker assumptions.

Numerical experiments confirm CGM's effectiveness.

Enhanced theoretical guarantees for CGM.

Abstract

The constrained gradient method (CGM) has recently been proposed to solve convex optimization and monotone variational inequality (VI) problems with general functional constraints. While existing literature has established convergence results for CGM, the assumptions employed therein are quite restrictive; in some cases, certain assumptions are mutually inconsistent, leading to gaps in the underlying analysis. This paper aims to derive rigorous and improved convergence guarantees for CGM under weaker and more reasonable assumptions, specifically in the context of strongly convex optimization and strongly monotone VI problems. Preliminary numerical experiments are provided to verify the validity of CGM and demonstrate its efficacy in addressing such problems.