Lagrange Duality in Set Optimization

TL;DR

This paper develops a new Lagrangian duality theory for set-valued optimization problems using a complete-lattice approach, extending scalar formulas and establishing strong duality under weak conditions.

Contribution

It introduces a novel duality framework for set optimization, including saddle sets and weak assumptions for strong duality.

Findings

Established a strong duality theorem with dual solutions.

Introduced saddle sets as a generalization of saddle points.

Extended scalar duality formulas to set-valued contexts.

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.