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1997, Journal of Mathematical Analysis and Applications
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.
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.
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.
Journal of Mathematical Analysis and Applications, 2007
The aim of this paper is to extend the so-called perturbation approach in order to deal with conjugate duality for constrained vector optimization problems. To this end we use two conjugacy notions introduced in the past in the literature in the framework of set-valued optimization. As a particular case we consider a vector variational inequality which we rewrite in the form of a vector optimization problem. The conjugate vector duals introduced in the first part allow us to introduce new gap functions for the vector variational inequality. The properties in the definition of the gap functions are verified by using the weak and strong duality theorems.
Journal of Mathematical Analysis and Applications - J MATH ANAL APPL, 1992
Journal of Mathematical Analysis and Applications, 2007
The aim of this work is to obtain scalar representations of set-valued optimization problems without any convexity assumption. Using a criterion of solution introduced by Kuroiwa [D. Kuroiwa, Some duality theorems of set-valued optimization with natural criteria, in:
Journal of Optimization Theory and Applications, 1995
In this paper, we are concerned with a bilevel multiobjective optimization problem (P). Using the function introduced by Gadhi and Dempe [Necessary optimality conditions and a new approach to multiobjective bilevel optimization problems. J Optim Theory Appl. 2012;155:100-114], we reformulate (P) as a single level mathematical programming problem (P *) and establish/exhibit the global equivalence between the two problems (P) and (P *). Using a generalized convexity introduced by Dutta and Chandra [Convexificator, generalized convexity and vector optimization. Optimization. 2004;53:77-94], we derive sufficient optimality conditions for the problem (P) and establish Mond-Weir duality results. To illustrate the obtained results some examples are given.
Journal of Convex Analysis
In this paper, we extend the notion of cone-subconvexlikeness of set-valued maps on topological linear spaces to set-valued maps on linear spaces, (that is, general linear spaces without any particular topology), and we provide several characterizations. An alternative theorem is also established for this kind of maps. Using the notion of vector closure introduced recently by Adán and Novo, we also provide, in this framework, an adaptation of the proper efficiency in the sense of Benson for set-valued maps. The previous notion and results are then applied to obtain optimality conditions of weak efficiency and a characterization of Benson proper efficiency by means of scalarization and multipliers rules.
Operations Research Forum
In this paper, a class of E-differentiable vector optimization problems with both inequality and equality constraints is considered. The so-called vector mixed E-dual problem is defined for the considered E-differentiable vector optimization problem with both inequality and equality constraints. Then, several mixed E-duality theorems are established under (generalized) V-E-invexity hypotheses.
The authors have previously shown some properties of scalarizing functions for set-valued maps. Fan’s inequality for set-valued maps was proved by using those properties. In this paper, we present new inherited properties which are obtained by generalized convexity and cone-semicontinuity assumptions for set-valued maps. By applying those new properties to Fan’s inequality for set-valued maps, we prove existence theorems for generalized vector equilibrium problems.
Journal of Mathematical Analysis and Applications, 2011
In this article we present a conjugate duality for a problem of maximizing a polyhedral concave nondecreasing homogeneous function over a convex feasible set in the nonnegative n-dimensional orthant. Using this duality we obtain a zero-gap duality for a vectormaximization problem.
Journal of the Australian Mathematical Society, 2003
The main aim of this paper is to obtain optimality conditions for a constrained set-valued optimization problem. The concept of Clarke epiderivative is introduced and is used to derive necessary optimality conditions. In order to establish sufficient optimality criteria we introduce a new class of set-valued maps which extends the class of convex set-valued maps and is different from the class of invex set-valued maps.
Computers & Mathematics with Applications, 2010
We consider two criteria of a solution associated with a set-valued optimization problem, a vector criterion and a set criterion. We show how solutions of a vector type can help to find solutions of a set type and reciprocally. As an application, we obtain a sufficient condition for the existence of solutions of a set type via vector optimization.
Top, 2005
Usually, finite dimensional linear spaces, locally convex topological linear spaces or normed spaces are the framework for vector and multiobjective optimization problems. Likewise, several generalizations of convexity are used in order to obtain new results. In this paper we show several Lagrangian type duality theorems and saddle-points theorems. From these, we obtain some characterizations of several efficient solutions of vector optimization problems (VOP), such as weak and proper efficient solutions in Benson's sense. These theorems are generalizations of preceding results in two ways. Firstly, because we consider real linear spaces without any particular topology, and secondly because we work with a recently appeared convexlike type of convexity. This new type, designated GVCL in this paper, is based on a new algebraic closure which we named vector closure.
Mathematics
In this paper, by using the normal subdifferential and equilibrium-like function we first obtain some properties for K-preinvex set-valued maps. Secondly, in terms of this equilibrium-like function, we establish some sufficient conditions for the existence of super minimal points of a K-preinvex set-valued map, that is, super efficient solutions of a set-valued vector optimization problem, and also attain necessity optimality terms for a general type of super efficiency.
HAL (Le Centre pour la Communication Scientifique Directe), 2014
The series in Vector Optimization contains publications in various fields of optimization with vector-valued objective functions, such as multiobjective optimization, multi criteria decision making, set optimization, vector-valued game theory and border areas to financial mathematics, biosystems, semidefinite programming and multiobjective control theory. Studies of continuous, discrete, combinatorial and stochastic multiobjective models in interesting fields of operations research are also included. The series covers mathematical theory, methods and applications in economics and engineering. These publications being written in English are primarily monographs and multiple author works containing current advances in these fields.
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.
2021
In this paper, we deal with optimization problems without assuming any topology. We study approximate efficiency and Q-Henig proper efficiency for the setvalued vector optimization problems, where Q is not necessarily convex. We use scalarization approaches based on nonconvex separation function to present some necessary and sufficient conditions for approximate (proper and weak) efficient solutions.
Computers & Mathematics with Applications, 2008
In this paper we study necessary and sufficient optimality conditions for a set-valued optimization problem. Convexity of the multifunction and the domain is not required. A definition of K-approximating multifunction is introduced. This multifunction is the differentiability notion applied to the problem. A characterization of weak minimizers is obtained for invex and generalized K-convexlike multifunctions using the Lagrange multiplier rule.
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.
2019
In this paper, a class of E-differentiable multiobjective programming problems with both inequality and equality constraints is considered. The so-called vector mixed E-dual problem is defined for the considered E-differentiable multiobjective programming problem with both inequality and equality constraints. Then, several mixed E-duality theorems are established under (generalized) E-convexity hypotheses. Further, so-called vector Mond-Weir E-dual and vector Wolfe E-dual problems are also defined for the considered E-differentiable multiobjective programming problem as special cases of its vector mixed E-dual problem.
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