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

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.