Existence results on Lagrange multiplier approach for gradient flows and application to optimization

TL;DR

This paper establishes existence and uniqueness results for the Lagrange multiplier approach in gradient flows, demonstrating its theoretical validity and potential application in optimization.

Contribution

It provides the first rigorous existence and uniqueness proofs for the Lagrange multiplier method in gradient flows, extending its applicability to optimization problems.

Findings

Existence of solutions under mild assumptions

Uniqueness and concrete bounds in special cases

Application to optimization problems

Abstract

This paper deals with the geometric numerical integration of gradient flow and its application to optimization. Gradient flows often appear as model equations of various physical phenomena, and their dissipation laws are essential. Therefore, dissipative numerical methods, which are numerical methods replicating the dissipation law, have been studied in the literature. Recently, Cheng, Liu, and Shen proposed a novel dissipative method, the Lagrange multiplier approach, for gradient flows, which is computationally cheaper than existing dissipative methods. Although their efficacy is numerically confirmed in existing studies, the existence results of the Lagrange multiplier approach are not known in the literature. In this paper, we establish some existence results. We prove the existence of the solution under a relatively mild assumption. In addition, by restricting ourselves to a special case, we show some existence and uniqueness results with concrete bounds. As gradient flows also appear in optimization, we further apply the latter results to optimization problems.