Augmented Lagrangian methods for infeasible convex optimization problems and diverging proximal-point algorithms

TL;DR

This paper analyzes the convergence of augmented Lagrangian methods for infeasible convex problems, showing they converge to solutions of the closest feasible problem and exploring their behavior under various assumptions.

Contribution

It provides new convergence results for ALMs applied to infeasible problems and introduces technical insights into proximal-point algorithms without minimizers.

Findings

ALMs converge to the closest feasible solution under mild conditions

Established convergence rates for ALMs in infeasible settings

Provided new results on proximal-point algorithms for non-minimizing functions

Abstract

This work investigates the convergence behavior of augmented Lagrangian methods (ALMs) when applied to convex optimization problems that may be infeasible. ALMs are a popular class of algorithms for solving constrained optimization problems. We demonstrate that, under mild assumptions, the sequences of iterates generated by ALMs converge to solutions of the ``closest feasible problem''. We establish progressively stronger convergence results, ranging from basic sequence convergence to more precise convergence rates, under a hierarchy of assumptions. This study leverages the classical relationship between ALMs and the proximal-point algorithm applied to the dual problem. A key technical contribution is a set of concise results on the behavior of the proximal-point algorithm when applied to functions that may lack minimizers. These results pertain to its convergence in terms of its subgradients and of the values of the convex conjugate.