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Characterisation of zero duality gap for optimization problems in spaces without linear structure

Ewa Bednarczuk, Monika Syga

Abstract

We prove sufficient and necessary conditions ensuring zero duality gap for Lagrangian duality in some classes of nonconvex optimization problems. To this aim, we use the $Φ$-convexity theory and minimax theorems for $Φ$-convex functions. The obtained zero duality results apply to optimization problems involving prox-bounded functions, DC functions, weakly convex functions and paraconvex functions as well as infinite-dimensional linear optimization problems, including Kantorovich duality which plays an important role in determining Wasserstein distance.

Characterisation of zero duality gap for optimization problems in spaces without linear structure

Abstract

We prove sufficient and necessary conditions ensuring zero duality gap for Lagrangian duality in some classes of nonconvex optimization problems. To this aim, we use the -convexity theory and minimax theorems for -convex functions. The obtained zero duality results apply to optimization problems involving prox-bounded functions, DC functions, weakly convex functions and paraconvex functions as well as infinite-dimensional linear optimization problems, including Kantorovich duality which plays an important role in determining Wasserstein distance.
Paper Structure (13 sections, 13 theorems, 128 equations)

This paper contains 13 sections, 13 theorems, 128 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $\Phi$-conjugation
  4. Minimax Theorem
  5. Lagrangian duality
  6. The $\Psi$-convexity of optimal value function
  7. Zero duality gap via intersection property
  8. Intersection property and the lower semicontinuity of optimal value function
  9. Examples
  10. Class $\Psi_{d}$
  11. Class $\Psi_{lsc}$
  12. Infinite-dimensional conic linear programming
  13. Kantorovich duality

Key Result

Theorem 2.3

(rolewicz and Theorem 1.2.6, rubbook, Theorem 7.1) Function $f:X\rightarrow(-\infty,+\infty]$ is $\Phi$-convex if and only if

Theorems & Definitions (24)

  • Definition 2.1
  • Example 2.2
  • Theorem 2.3
  • Definition 2.4
  • Lemma 2.5
  • Theorem 2.6
  • Remark 2.7
  • Proposition 3.1
  • Proposition 3.2
  • Remark 3.3
  • ...and 14 more