On the Convergence of Constrained Gradient Method

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 derives rigorous and improved convergence guarantees for CGM under weaker and more reasonable assumptions, covering strongly convex optimization problems, as well as VI problems under both strongly monotone and general monotone settings. Preliminary numerical experiments are provided to verify the validity of CGM and demonstrate its efficacy in addressing such problems.