Remarks on the $\Gamma $-regularization of Non-convex and Non-semi-continuous Functions on Topological Vector Spaces

TL;DR

This paper introduces a method using $ ext{Gamma}$-regularization to analyze minimization problems of non-convex, non-semi-continuous functions on topological vector spaces, extending known theorems to broader contexts.

Contribution

It presents a novel approach to study non-convex functions via convex regularization and generalizes the Lanford III-Robinson theorem beyond separable Banach spaces.

Findings

Establishes $ ext{Gamma}$-regularization as a key tool for non-convex minimization.

Generalizes the Lanford III-Robinson theorem to all locally convex real spaces.

Provides a new framework for understanding subdifferentials of non-convex functions.

Abstract

We show that the minimization problem of any non-convex and non-lower semi-continuous function on a compact convex subset of a locally convex real topological vector space can be studied via an associated convex and lower semi-continuous function $\Gamma \left( h\right) $. This observation uses the notion of $\Gamma $-regularization as a key ingredient. As an application we obtain, on any locally convex real space, a generalization of the Lanford III-Robinson theorem which has only been proven for separable real Banach spaces. The latter is a characterization of subdifferentials of convex continuous functions.