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On Fenchel c-conjugate dual problems for DC optimization: characterizing weak, strong and stable strong duality

M. D. Fajardo, J. Vidal-Nunez

TL;DR

This work introduces two Fenchel-type duals for DC optimization via $c$-conjugation, clarifying when weak, strong, and stable strong duality hold within the framework of evenly convex functions. It provides necessary and sufficient conditions for duality gaps expressed through epigraphs and conjugates, and it develops an $e$-convexification technique to generate an auxiliary dual that brackets the primal value when $g$ is $e$-convex. The paper also extends the analysis to stable strong duality under linear perturbations and offers a unifying perspective by linking the duality properties to $e'$-convex hulls. Overall, the results deepen the regularity theory for DC problems beyond classical convex settings and suggest perturbation-based avenues for further refinement.

Abstract

In this paper we present two Fenchel-type dual problems for a DC (difference of convex functions) optimization primal one. They have been built by means of the c-conjugation scheme, a pattern of conjugation which has been shown to be suitable for evenly convex functions. We study characterizations of weak, strong and stable strong duality for both pairs of primal-dual problems. We also give conditions which relate the existence of strong and stable strong duality for both pairs.

On Fenchel c-conjugate dual problems for DC optimization: characterizing weak, strong and stable strong duality

TL;DR

This work introduces two Fenchel-type duals for DC optimization via -conjugation, clarifying when weak, strong, and stable strong duality hold within the framework of evenly convex functions. It provides necessary and sufficient conditions for duality gaps expressed through epigraphs and conjugates, and it develops an -convexification technique to generate an auxiliary dual that brackets the primal value when is -convex. The paper also extends the analysis to stable strong duality under linear perturbations and offers a unifying perspective by linking the duality properties to -convex hulls. Overall, the results deepen the regularity theory for DC problems beyond classical convex settings and suggest perturbation-based avenues for further refinement.

Abstract

In this paper we present two Fenchel-type dual problems for a DC (difference of convex functions) optimization primal one. They have been built by means of the c-conjugation scheme, a pattern of conjugation which has been shown to be suitable for evenly convex functions. We study characterizations of weak, strong and stable strong duality for both pairs of primal-dual problems. We also give conditions which relate the existence of strong and stable strong duality for both pairs.
Paper Structure (8 sections, 16 theorems, 64 equations, 3 figures)

This paper contains 8 sections, 16 theorems, 64 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Dual problems
  4. E-convexification technique to get another dual problem
  5. Conditions for weak duality and zero duality gap in DC problems
  6. Conditions for strong duality in DC problems
  7. Stable Strong Duality
  8. Conclusions

Key Result

Theorem 2.3

Let $f:X\rightarrow \overline{\mathbb{R}}$ and $g:W\rightarrow \overline{\mathbb{R}}$. Then

Figures (3)

  • Figure 1: Set in Example \ref{['ex:Econvex_non_closed']}
  • Figure 2: Epigraphs of the functions $f$ and $g$ in Example \ref{['ex:Example_g_non_econvex_and_WD']}
  • Figure 3: Projections onto $\mathbb{R}^2$ in Example \ref{['ex:Strict_containments']}

Theorems & Definitions (23)

  • Definition 2.1: Def. 1, DML2002
  • Example 2.2
  • Theorem 2.3: Prop. 6.1, 6.2, Cor. 6.1, ML2005
  • Example 3.1
  • Lemma 4.1
  • Remark 1
  • Proposition 4.2
  • Proposition 4.3
  • Proposition 5.1
  • Example 5.2
  • ...and 13 more