Continuous dynamics related to monotone inclusions and non-smooth optimization problems

TL;DR

This survey reviews variational analysis techniques and dynamical systems for solving monotone inclusions and non-smooth optimization problems, highlighting the use of Lyapunov theory and extending to non-convex cases with Kurdyka-Lojasiewicz property.

Contribution

It provides a comprehensive overview of first and second order dynamical systems for monotone and non-smooth optimization, including recent results for non-convex problems.

Findings

Lyapunov functions are key for asymptotic analysis.

Resolvent and proximal operators are central tools.

Results extend to non-convex optimization with Kurdyka-Lojasiewicz property.

Abstract

The aim of this survey is to present the main important techniques and tools from variational analysis used for first and second order dynamical systems of implicit type for solving monotone inclusions and non-smooth optimization problems. The differential equations are expressed by means of the resolvent (in case of a maximally monotone set valued operator) or the proximal operator for non-smooth functions. The asymptotic analysis of the trajectories generated relies on Lyapunov theory, where the appropriate energy functional plays a decisive role. While the most part of the paper is related to monotone inclusions and convex optimization problems in the variational case, we present also results for dynamical systems for solving non-convex optimization problems, where the Kurdyka-Lojasiewicz property is used.