An Augmented Lagrangian Method for Optimization Problems in Banach Spaces

TL;DR

This paper introduces a novel augmented Lagrangian method tailored for constrained optimization in Banach spaces, capable of handling infinite-dimensional inequalities without convexity or second-order assumptions, with proven convergence and practical demonstrations.

Contribution

It develops a new augmented Lagrangian approach for Banach space optimization that relaxes traditional assumptions and extends to infinite-dimensional inequality constraints.

Findings

The method converges under broad conditions.

It effectively handles infinite-dimensional inequality constraints.

Numerical results demonstrate practical applicability.

Abstract

We propose a variant of the classical augmented Lagrangian method for constrained optimization problems in Banach spaces. Our theoretical framework does not require any convexity or second-order assumptions and allows the treatment of inequality constraints with infinite-dimensional image space. Moreover, we discuss the convergence properties of our algorithm with regard to feasibility, global optimality, and KKT conditions. Some numerical results are given to illustrate the practical viability of the method.