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Convex analysis for composite functions without K-convexity

Juan Pablo Vielma

TL;DR

The paper studies convex analysis of composite functions without relying on $K$-convexity, by leveraging perturbation-based Lagrangian duality to derive conjugate and sub-differential formulas for general composites $f(x,u)=f_0(x)+g(F(x)+u)$. It shows that many results traditionally tied to $K$-convexity can be recovered under elemental convex analysis, and provides less restrictive necessary conditions for conjugates and sub-differentials. A detailed comparison with $K$-convexity based results highlights that while $K$-convexity aids verification, it is not essential for the convex analysis of composite functions. The framework also covers additive and GNLP extensions, offering structured duality conditions, saddle-point characterizations, and explicit chain rules with proofs deferred to appendices. Overall, the work improves accessibility and applicability of composite-function convex analysis, especially in finite-dimensional settings, by reducing reliance on $K$-convexity and clarifying necessary conditions for conjugate and sub-differential formulas.

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.

Convex analysis for composite functions without K-convexity

TL;DR

The paper studies convex analysis of composite functions without relying on -convexity, by leveraging perturbation-based Lagrangian duality to derive conjugate and sub-differential formulas for general composites . It shows that many results traditionally tied to -convexity can be recovered under elemental convex analysis, and provides less restrictive necessary conditions for conjugates and sub-differentials. A detailed comparison with -convexity based results highlights that while -convexity aids verification, it is not essential for the convex analysis of composite functions. The framework also covers additive and GNLP extensions, offering structured duality conditions, saddle-point characterizations, and explicit chain rules with proofs deferred to appendices. Overall, the work improves accessibility and applicability of composite-function convex analysis, especially in finite-dimensional settings, by reducing reliance on -convexity and clarifying necessary conditions for conjugate and sub-differential formulas.

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.
Paper Structure (24 sections, 37 theorems, 122 equations)

This paper contains 24 sections, 37 theorems, 122 equations.

Table of Contents

  1. Introduction
  2. Notation and assumptions
  3. Perturbation based Lagrangian duality
  4. General weak duality and optimality conditions
  5. Convex strong duality
  6. Generalized nonlinear programming
  7. Composite functions
  8. Sub-differential chain rule
  9. Combining with standard additive results
  10. Connection of Theorem \ref{['simplecomposite']} to K-convexity
  11. Comparison between Theorems \ref{['simplecomposite']} and \ref{['burketheo']}
  12. Using K-convexity to satisfy necessary conditions
  13. Additive composite functions
  14. Omitted Proofs
  15. Proofs from Section \ref{['convex:comp:section']}
  16. ...and 9 more sections

Key Result

Theorem 2.1

For any $f:\mathbb{R}^n\times \mathbb{R}^m\to \overline{\mathbb{R}}$ and $(\bar{v},\bar{u})\in \mathbb{R}^n\times \mathbb{R}^m$ we have that $q_{\bar{u}}$ is convex, and $q_{\bar{u}}(\bar{v}):=\inf_{y\in \mathbb{R}^m}\sup_{x\in \mathbb{R}^n}\left\{\langle\bar{v}, x\rangle-\langle\bar{u}, y\rangle -l(x,y) \right\}$. In addition, Finally, for any $(\bar{v},\bar{u})\in \mathbb{R}^n\times \mathbb{R

Theorems & Definitions (70)

  • Definition 1.1
  • Definition 2.1
  • Theorem 2.1
  • Proposition 2.1
  • Theorem 2.2
  • Definition 2.2
  • Corollary 2.1
  • Corollary 2.2
  • Lemma 2.1: rockafellar2023augmented
  • Definition 3.1
  • ...and 60 more