A Perturbation Approach to Vector Optimization Problems: Lagrange and Fenchel-Lagrange Duality

TL;DR

This paper develops a perturbation approach to vector optimization problems, establishing duality results including Lagrange and Fenchel-Lagrange duality, with applications to various classes of cone-constrained vector problems.

Contribution

It introduces new perturbation mappings and duality frameworks for vector optimization, including stable strong duality results and novel Fenchel-Lagrange duality theorems.

Findings

Established representations of conjugate epigraphs

Proved vector Farkas lemmas and duality theorems

Derived new Fenchel-Lagrange duality results for vector problems

Abstract

In this paper we study the general minimization vector problem (P), concerning a perturbation mapping, defined in locally convex Hausdorff topological vector spaces where the "WInf" stands for the weak infimum with respect to an ordering generated by a convex cone $K$. Several representations of the epigraph of the conjugate mapping of the perturbation mapping are established. From these, variants vector Farkas lemmas are then proved. Armed with these basic tools, the {\it dual} and the so-called {\it loose dual problem} of (P) are defined, and then stable strong duality results between these pairs of primal-dual problems are established. The results just obtained are then applied to a general class (CCCV) of composed vector optimization problems with cone-constrained. For this classes of problems, four perturbation mappings are suggested. Each of these mappings yields several forms of vector Farkas lemmas and two forms of dual problems for (CCVP). Concretely, one of the suggested perturbation mapping give rises to well-known {\it Lagrange} and {\it loose Lagrange dual problems} for (CCVP) while each of the three others, yields two kinds of Fenchel-Lagrange dual problems for (CCVP). Stable strong duality for these pairs of primal-dual problems are proved. Several special cases of (CCVP) are also considered at the end of the paper, including: vector composite problems (without constraints), cone-constrained vector problems, and scalar composed problems. The results obtained in this papers when specified to the two concrete mentioned vector problems go some Lagrange duality results appeared recently, and also lead to new results on stable strong Fenchel-Lagrange duality results, which, to the best knowledge of the authors, appear for the first time in the literature.