Conjugate duality in vector optimization

Arxiv preprint arXiv:1112.1315, 2011

Over the past years a theory of conjugate duality for set-valued functions that map into the set of upper closed subsets of a preordered topological vector space was developed. For scalar duality theory, continuity of convex functions plays an important role. For set-valued maps different notions of continuity exist. We will compare the most prevalent ones in the special case that the image space is the set of upper closed subsets of a preordered topological vector space and analyze which of the results can be conveyed from the extended real-valued case.

Optimization, 2013

We define the quasi-minimal elements of a set with respect to a convex cone and characterize them via linear scalarization. Then we attach to a general vector optimization problem a dual vector optimization problem with respect to quasi-efficient solutions and establish new duality results. By considering particular cases of the primal vector optimization problem we derive vector dual problems with respect to quasi-efficient solutions for both constrained and unconstrained vector optimization problems and the corresponding weak, strong and converse duality statements.

We consider the classical duality operators for convex objects such as the polar of a convex set containing the origin, the dual norm, the Fenchel-transform of a convex function and the conjugate of a convex cone. We give a new, sharper, unified treatment of the theory of these operators, deriving generalized theorems of Hahn-Banach, Fenchel-Moreau and Dubovitsky-Milyutin for the conjugate of convex cones in not necessarily finite dimensional vector spaces and hence for all the other duality operators of convex objects.

Journal of Mathematical Analysis and Applications, 1997

In this paper, conjugate duality results for convexlike set-valued vector optimization problems are presented under closedness or boundedness hypotheses. Some properties of the value mapping of a set-valued vector optimization problem are studied. A conjugate duality result is also proved for a convex set-valued vector optimization problem without the requirements of closedness and boundedness.

Applied Mathematics Letters, 1999

Journal of Mathematical Analysis and Applications, 1996

In this note, a general cone separation theorem between two subsets of image space is presented. With the aid of this, optimality conditions and duality for vector optimization of set-valued functions in locally convex spaces are discussed.

Journal of Mathematical Analysis and Applications, 1974

Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 2014

Given a convex optimization problem (P) in a locally convex topological vector space X with an arbitrary number of constraints, we consider three possible dual problems of (P) ; namely, the usual Lagrangian dual (D) ; the perturbational dual (Q) ; and the surrogate dual () ; the last one recently introduced in a previous paper of the authors [M.A. Goberna, M.A. López, M. Volle, Primal attainment in convex in…nite optimization duality, J. Convex Analysis 21, No. 4, 2014, in press]. As shown by simple examples, these dual problems may be all di¤erent. This paper provides conditions ensuring that inf(P) = max(D); inf(P) = max(Q); and inf(P) = max () (dual equality and existence of dual optimal solutions) in terms of the so-called closedness regarding to a set. Su¢ cient conditions guaranteeing min(P) = sup(Q) (dual equality and existence of primal optimal solutions) are also provided, for the nominal problems and also for their perturbational relatives. The particular cases of convex semi-in…nite optimization problems (in which either the number of constraints or the dimension of X, but not both, is …nite) and linear in…nite optimization problems are analyzed. Finally, some applications to the feasibility of convex inequality systems are described.

Commun. Appl. Anal, 2009

ABSTRACT. We prove a strong duality result between a convex optimization problem with both cone and equality constraints and its Lagrange dual formulation, provided that a constraint qualification condition related to the notion of quasi-relative interior holds true. ...

Optimization, 2022

In this paper we concern the vector problem of the model: (VP) WInf{F (x) : x ∈ C, G(x) ∈ −S}. where X, Y, Z are locally convex Hausdorff topological vector spaces, F : X → Y ∪ {+∞Y } and G : X → Z ∪ {+∞Z} 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 establish 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 dual problem and two new kinds of Fenchel-Lagrange dual problems for the vector problem (VP). Strong and stable strong duality results corresponding to these three mentioned dual problems of (VP) are established with the help of new Farkas-type results just obtained from the representations. It is shown that in the special case where Y = R, the Lagrange and Fenchel-Lagrange dual problems for (VP), go back to Lagrange dual problem, and Fenchel-Lagrange dual problems for scalar problems, and the resulting duality results cover, and in some setting, extend the corresponding ones for scalar problems in the literature.