Lagrange Multipliers, Duality, and Sensitivity in Set-Valued Convex Programming via Pointed Closed Convex Processes

TL;DR

This work develops a novel set-valued Lagrangian duality theory for convex optimization problems where both the objective and constraints are set-valued maps between preordered normed spaces. By introducing a Lagrange multiplier theorem that yields dual variables as pointed closed convex processes, and by formulating a corresponding dual program with a strong duality relation, the authors connect dual solutions to the sensitivity of the primal problem. The analysis hinges on nondominated points and Slater-type conditions, and is complemented by a detailed sensitivity framework using adjoint processes and contingent derivatives. Overall, the approach extends scalar Lagrange duality methods to the set-valued vector setting and clarifies how dual information governs primal behavior, with potential extensions to non-convex settings and open questions about multiplier selection.

Abstract

We present a new kind of Lagrangian duality theory for set-valued convex optimization problems whose objective and constraint maps are defined between preordered normed spaces. The theory is accomplished by introducing a new set-valued Lagrange multiplier theorem and a dual program with variables that are pointed closed convex processes. The pointed nature assumed for the processes is essential for the derivation of the main results presented in this research. We also develop a strong duality theorem that guarantees the existence of dual solutions, which are closely related to the sensitivity of the primal program. It allows extending the common methods used in the study of scalar programs to the set-valued vector case.

Theorems & Definitions (28)

• Definition 3.1
• Remark 1
• Definition 3.2
• Definition 3.3
• Remark 2
• Definition 3.5
• Lemma 3.6
• Lemma 3.7
• Lemma 3.8
• Theorem 3.9