Lagrange Duality in Set Optimization

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.

References in corpus (4)

• An Algorithm to Solve Polyhedral Convex Set Optimization Problems
• Lagrange duality, stability and subdifferentials in vector optimization
• Directional derivatives and subdifferentials of set-valued convex functions
• Continuity of Convex Set-valued Maps and a Fundamental Duality Formula for Set-valued Optimization

Cited by in corpus (12)

• Set optimization - a rather short introduction
• Primal and Dual Approximation Algorithms for Convex Vector Optimization Problems
• Set-valued shortfall and divergence risk measures
• An Algorithm to Solve Polyhedral Convex Set Optimization Problems
• Lagrange duality, stability and subdifferentials in vector optimization
• Set Relations and Approximate Solutions in Set Optimization
• Vector duality via conditional extension of dual pairs
• A Fenchel-Moreau theorem for ${\bar{L}}^0$-valued functions
• A Set-Valued Lagrange Theorem based on a Process for Convex Vector Programming
• Multistage Portfolio Optimization: A Duality Result in Conic Market Models
• Lagrange Multipliers, Duality, and Sensitivity in Set-Valued Convex Programming via Pointed Closed Convex Processes
• Complete Duality for Quasiconvex and Convex Set-Valued Functions