Convex analysis for composite functions without K-convexity

TL;DR

This paper extends convex analysis of composite functions beyond K-convexity assumptions, providing new formulas and conditions that are less restrictive and more broadly applicable in optimization problems.

Contribution

It introduces a framework that recovers and extends existing convex analysis results without relying on K-convexity, enhancing accessibility and applicability.

Findings

Standard duality methods can replace K-convexity assumptions.

New necessary conditions for conjugate and sub-differential formulas.

Comparison shows broader applicability of the new results.

Abstract

Composite functions have been studied for over 40 years and appear in a wide range of optimization problems. Convex analysis of these functions focuses on (i) conditions for convexity of the function based on properties of its components, (ii) formulas for the convex conjugate of the function based on those of its components and (iii) chain-rule-like formulas for the sub-differential of the function. To the best of our knowledge all existing results on this matter are based on the notion of K-convexity of functions where K is a closed convex cone. These notions can be considered elementary when K is the non-negative orthant, but otherwise may limit the accessibility of the associated results. In this work we show how standard results on perturbation function based duality can be used to recover and extend existing results without the need for K-convexity. We also provide a detailed comparison with K-convexity based results and show that, while K-convexity is not needed for the convex analysis of composite functions, it certainly aids in verifying the conditions associated with our proposed results. Finally, we provide necessary conditions for the conjugate and sub-differential formulas that are less restrictive than those required by the existing results.