New representations of epigraphs of conjugate mappings and Lagrange, Fenchel-Lagrange duality for vector optimization problems

• Published in: Optimization 2023-06, Vol.72 (6), p.1429-1461
• Main Authors: Dinh, N., Long, D. H.
• Format: Article
• Language: English
• Subjects: (stable) vector Farkas lemmas; Conjugates; extended epigraphs of conjugate mappings; Lagrange and Fenchel-Lagrange duality for vector optimization problems; Optimization; qualification conditions; Representations; Vector inequalities; Vector spaces
• ISSN: 0233-1934; 1029-4945
• DOI: 10.1080/02331934.2021.2017431

In this paper, we concern the vector problem: where X, Y, Z are locally convex Hausdorff topological vector spaces, and are proper mappings, C is a nonempty convex subset of X, and S is a non-empty closed, convex cone in Z. Several new presentations of epigraphs of composite conjugate mappings associated to (VP) are established under variant qualifying conditions. The significance of these representations is twofold: Firstly, they play a key role in establishing new kinds of vector Farkas lemmas which serve as tools in the study of vector optimization problems; secondly, they pay the way to define Lagrange and two new kinds of Fenchel-Lagrange dual problems for (VP). Strong and stable strong duality results corresponding to these mentioned dual problems of (VP) are established using the new Farkas-type results just obtained. It is shown that in the special case where , the Lagrange and Fenchel-Lagrange dual problems for (VP), go back to Lagrange, and Fenchel-Lagrange dual problems for scalar problems, and the resulting duality results cover, and in some cases, extend the corresponding ones for scalar problems in the literature.