Exact augmented Lagrangians for constrained optimization problems in Hilbert spaces I: Theory

TL;DR

This paper develops a comprehensive theoretical framework for exact augmented Lagrangians in Hilbert space constrained optimization, highlighting their differentiability and advantages over nonsmooth penalty functions for numerical applications.

Contribution

It introduces a new class of smooth, exact augmented Lagrangians with theoretical properties and conditions linking KKT points to optimal solutions in Hilbert space problems.

Findings

Derived estimates for augmented Lagrangian and its gradient

Established conditions for KKT points to be local/global minimizers

Demonstrated applicability to variational, PDE, and optimal control problems

Abstract

In this two-part study, we develop a general theory of the so-called exact augmented Lagrangians for constrained optimization problems in Hilbert spaces. In contrast to traditional nonsmooth exact penalty functions, these augmented Lagrangians are continuously differentiable for smooth problems and do not suffer from the Maratos effect, which makes them especially appealing for applications in numerical optimization. Our aim is to present a detailed study of various theoretical properties of exact augmented Lagrangians and discuss several applications of these functions to constrained variational problems, problems with PDE constraints, and optimal control problems. The first paper is devoted to a theoretical analysis of an exact augmented Lagrangian for optimization problems in Hilbert spaces. We obtain several useful estimates of this augmented Lagrangian and its gradient, and present several types of sufficient conditions for KKT-points of a constrained problem corresponding to locally/globally optimal solutions to be local/global minimisers of the exact augmented Lagrangian.