Global optimization, the Gaussian ensemble, and universal ensemble equivalence

TL;DR

This paper explores conditions under which constrained optimization problems can be transformed into unconstrained ones with identical solutions, using concepts from statistical mechanics and ensemble theory.

Contribution

It introduces the Gaussian ensemble as a tool to establish universal ensemble equivalence in the context of global optimization.

Findings

Conditions for ensemble equivalence are characterized using large deviations and convex analysis.

The Gaussian ensemble provides a universal framework for relating constrained and unconstrained problems.

The approach bridges optimization and statistical mechanics, offering new insights into problem equivalence.

Abstract

Shortened abstract: Given a constrained minimization problem, under what conditions does there exist a related, unconstrained problem having the same minimum points? This basic question in global optimization motivates this paper, which answers it from the viewpoint of statistical mechanics. In this context, it reduces to the fundamental question of the equivalence and nonequivalence of ensembles, which is analyzed using the theory of large deviations and the theory of convex functions.