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

Fernando García-Castaño,Miguel Ángel Melguizo-Padial

doi:10.1007/s10957-022-02005-2

Fernando García-Castaño, Miguel Ángel Melguizo-Padial

Open Access

https://doi.org/10.1007/s10957-022-02005-2

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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.

Highlights

Set-valued optimization is an expanding branch in applied mathematics that has attracted a great deal of attention in the last decades [1, 12, 13, 22, 23]
This topic tackles optimization problems where the objective and/or the constraint maps are setvalued ones acting between abstract spaces
A closed convex process Δ : Z ⇒ Y is said to be a Lagrange multiplier of (P(0)) at y0, if y0 is a nondominated point of the program

Summary

Introduction

Set-valued optimization is an expanding branch in applied mathematics that has attracted a great deal of attention in the last decades [1, 12, 13, 22, 23]. The solution concept based on set approaches is based on a set order relation This is obtained by extending the original preordered image vector space to its power set. We deal with the solution concept based on a vector approach From this perspective, we establish a new Lagrange multiplier theorem. We prove that the sensitivity of the primal program is closely related to the set of dual solutions. Such solutions are usually continuous linear operators, the dual variables considered in this paper are pointed closed convex processes. The processes in this work are pointed This property seems to improve the adaptability of the variables to the structure of convex set-valued vector problems. 6, we present conclusions that summarize this work, and we pose some open problems for further research

Preliminaries and Notation

Problem Formulation and Set-Valued Lagrange Multipliers

Duality

Sensitivity Analysis

Conclusions