On Error Bounds and Multiplier Methods for Variational Problems in Banach Spaces

TL;DR

This paper characterizes local error bounds for variational problems in Banach spaces, proposes an augmented Lagrangian algorithm with convergence guarantees, and demonstrates its effectiveness through numerical experiments.

Contribution

It introduces a new characterization of local error bounds and analyzes an augmented Lagrangian method for solving general variational problems in Banach spaces.

Findings

Established conditions for local error bounds in Banach space variational problems.

Proved global convergence and convergence rate estimates for the proposed algorithm.

Presented numerical results demonstrating the method's applicability to control, equilibrium, and estimation problems.

Abstract

This paper deals with a general form of variational problems in Banach spaces which encompasses variational inequalities as well as minimization problems. We prove a characterization of local error bounds for the distance to the (primal-dual) solution set and give a sufficient condition for such an error bound to hold. In the second part of the paper, we consider an algorithm of augmented Lagrangian type for the solution of such variational problems. We give some global convergence properties of the method and then use the error bound theory to provide estimates for the rate of convergence and to deduce boundedness of the sequence of penalty parameters. Finally, numerical results for optimal control, Nash equilibrium problems, and elliptic parameter estimation problems are presented.