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Lagrange Duality in Set Optimization

Andreas H. Hamel, Andreas Löhne

TL;DR

Based on the complete-lattice approach, a new Lagrangian type duality theory for set-valued optimization problems is presented, and a strong duality theorem is given under very weak assumptions: the ordering cone may have an empty interior or may not be pointed.

Abstract

Based on the complete-lattice approach, a new Lagrangian duality theory for set-valued optimization problems is presented. In contrast to previous approaches, set-valued versions for the known scalar formulas involving infimum and supremum are obtained. In particular, a strong duality theorem, which includes the existence of the dual solution, is given under very weak assumptions: The ordering cone may have an empty interior or may not be pointed. "Saddle sets" replace the usual notion of saddle points for the Lagrangian, and this concept is proven to be sufficient to show the equivalence between the existence of primal/dual solutions and strong duality on the one hand and the existence of a saddle set for the Lagrangian on the other hand.

Lagrange Duality in Set Optimization

TL;DR

Based on the complete-lattice approach, a new Lagrangian type duality theory for set-valued optimization problems is presented, and a strong duality theorem is given under very weak assumptions: the ordering cone may have an empty interior or may not be pointed.

Abstract

Based on the complete-lattice approach, a new Lagrangian duality theory for set-valued optimization problems is presented. In contrast to previous approaches, set-valued versions for the known scalar formulas involving infimum and supremum are obtained. In particular, a strong duality theorem, which includes the existence of the dual solution, is given under very weak assumptions: The ordering cone may have an empty interior or may not be pointed. "Saddle sets" replace the usual notion of saddle points for the Lagrangian, and this concept is proven to be sufficient to show the equivalence between the existence of primal/dual solutions and strong duality on the one hand and the existence of a saddle set for the Lagrangian on the other hand.

Paper Structure

This paper contains 17 sections, 15 theorems, 118 equations.

Table of Contents

  1. Introduction
  2. Problem formulation, notation
  3. Solutions for set-valued optimization problems and canonical extensions
  4. Fenchel conjugates for set-valued functions
  5. Linear scalarization and the scalarized problems
  6. Scalarization of $\mathcal{G}(C)$-valued functions
  7. Scalarization of the optimization problem
  8. Duality
  9. Construction of the dual problem and weak duality
  10. Strong duality
  11. Strong duality via compactness
  12. An example
  13. Set-valued duality in vector optimization
  14. Canonical extensions and saddle points of Lagrangians
  15. Saddle points for functions with values in complete lattices
  16. ...and 2 more sections

Key Result

Proposition 2.2

(reconstruction of the primal) If $f\left( x \right) \in \mathcal{G}\left( C \right)$, $g\left( x \right) \in \mathcal{G}\left( D \right)$ and $f\left( x \right) \neq Z$ for each $x \in X$, then

Theorems & Definitions (27)

  • Remark 2.1
  • Proposition 2.2
  • Remark 2.3
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Remark 3.4
  • Example 3.5
  • Definition 3.6
  • Definition 3.7
  • ...and 17 more