On Lagrange multiplier theorems for non-smooth optimization for a large class of variational models in Banach spaces

TL;DR

This paper develops new optimality conditions for a broad class of non-smooth variational models in Banach spaces, using softer proof techniques that extend existing theoretical frameworks.

Contribution

It introduces novel proof procedures for Lagrange multiplier theorems in non-smooth optimization within Banach spaces, applicable to models with equality and inequality constraints.

Findings

Established optimality conditions for non-smooth variational models

Extended Lagrange multiplier theorems to more general Banach space settings

Provided softer proof techniques compared to traditional methods

Abstract

This article develops optimality conditions for a large class of non-smooth variational models. The main results are based on standard tools of functional analysis and calculus of variations. Firstly we address a model with equality constraints and, in a second step, a more general model with equality and inequality constraints, always in a general Banach space context. We highlight the results in general are well known, however, some novelties are introduced related to the proof procedures, which are in general softer than those concerning the present literature.