Conditional Infimum and Hidden Convexity in Optimization

TL;DR

This paper introduces the concept of conditional infimum to detect hidden convexity in nonconvex optimization problems, providing a theoretical framework and practical criteria for identifying convex structures within nonconvex problems.

Contribution

It develops the theory of conditional infimum, establishes a tower property, and offers a new sufficient condition for hidden convexity in nonconvex minimization problems.

Findings

Conditional infimum helps detect hidden convexity.

A tower property for the conditional infimum is established.

Application to nonconvex quadratic minimization problems.

Abstract

Detecting hidden convexity is one of the tools to address nonconvex minimization problems. After giving a formal definition of hidden convexity, we introduce the notion of conditional infimum, as it will prove instrumental in detecting hidden convexity. We develop the theory of the conditional infimum, and we establish a tower property, relevant for minimization problems. Thus equipped, we provide a sufficient condition for hidden convexity in nonconvex minimization problems. We illustrate our result on nonconvex quadratic minimization problems. We conclude with perspectives for using the conditional infimum in relation to the so-called S-procedure, to couplings and conjugacies, and to lower bound convex programs.