Conjugate Duality in Constrained Set-Valued Vector Optimization Problems

In this article, under a concept of supremum/infimum of a set, defined in terms of a closure of the set, three kinds of conjugate dual problems are proposed for a constrained set-valued vector optimization problem. Weak duality, strong duality, and stability criteria are investigated. The inclusion relations between the image sets of the dual problems are also discussed.     

On approximate solutions in set-valued optimization problems

In this paper we introduce several concepts of approximate solutions of set-valued optimization problems with vector and set optimization. We prove existence results and necessary and sufficient conditions by using limit sets.     

向量集值映射的共轭对偶

借助抽象算子将共轭映射的概念的到抽象空间,引入了集值映射的共轭映射和次梯度,据此讨论了集值映射共轭对偶的全局稳定性。     

Conjugate duality in vector optimization and some applications to the vector variational inequality

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.     

Set-valued increasing-along-rays maps and well-posed set-valued star-shaped optimization

In this paper we introduce a class of set-valued increasing-along-rays maps and present some properties of set-valued increasing-along-rays maps. We show that the increasing-along-rays property of a set-valued map is close related to the corresponding set-valued star-shaped optimization. By means of increasing-along-rays property, we investigate stability and well-posedness of set-valued star-shaped optimization.     

Conjugate duality for vector-maximization problems

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 vector-maximization problem.     

A new Fenchel dual problem in vector optimization

We introduce a new Fenchel dual for vector optimization problems inspired by the form of the Fenchel dual attached to the scalarized primal multiobjective problem. For the vector primal-dual pair we prove weak and strong duality. Furthermore, we recall two other Fenchel-type dual problems introduced in the past in the literature, in the vector case, and make a comparison among all three duals. Moreover, we show that their sets of maximal elements are equal.     

First-order optimality conditions in set-valued optimization

A a set-valued optimization problem min _C F(x), x ∈X ₀, is considered, where X ₀ ⊂ X, X and Y are normed spaces, F: X ₀ ⊂ Y is a set-valued function and C ⊂ Y is a closed cone. The solutions of the set-valued problem are defined as pairs (x ⁰,y ⁰), y ⁰ ∈F(x ⁰), and are called minimizers. The notions of w-minimizers (weakly efficient points), p-minimizers (properly efficient points) and i-minimizers (isolated minimizers) are introduced and characterized through the so called oriented distance. The relation between p-minimizers and i-minimizers under Lipschitz type conditions is investigated. The main purpose of the paper is to derive in terms of the Dini directional derivative first order necessary conditions and sufficient conditions a pair (x ⁰, y ⁰) to be a w-minimizer, and similarly to be a i-minimizer. The i-minimizers seem to be a new concept in set-valued optimization. For the case of w-minimizers some comparison with existing results is done.     

Conjugate Maps,Subgradients and Conjugate Duality in Set-Valued Optimization

Using the concept of supremum/infimum of a set, defined in terms of the closure of the set, we introduce the notions of conjugate and biconjugate maps as well as that of subgradients of a set-valued map. Conjugate duality results are also established for a set-valued optimization problem.     

Explicitly quasiconvex set-valued optimization

The principal aim of this paper is to extend some recent results concerning the contractibility of efficient sets and the Pareto reducibility in multicriteria explicitly quasiconvex optimization problems to similar vector optimization problems involving set-valued objective maps. To this end, an appropriate notion of generalized convexity is introduced for set-valued maps taking values in a partially ordered real linear space, which naturally extends the classical concept of explicit quasiconvexity of real-valued functions. Actually, the class of so-called explicitly cone-quasiconvex set-valued maps in particular contains the cone-convex set-valued maps, and it is contained in the class of cone-quasiconvex set-valued maps.      

On φ-strongly accretive mappings and some set-valued variational problems

This paper shows that a continuous φ-strongly accretive mapping on a real Banach space is single-valued. And some recent results of set-valued variational inclusions and inequalities are discussed.     

Pseudopower expansion of solutions of generalized equations and constrained optimization problems

We show that the solution of a strongly regular generalized equation subject to a scalar perturbation expands in pseudopower series in terms of the perturbation parameter, i.e., the expansion of orderk is the solution of generalized equations expanded to orderk and thus depends itself on the perturbation parameter. In the polyhedral case, this expansion reduces to a usual Taylor expansion. These results are applied to the problem of regular perturbation in constrained optimization. We show that, if the strong regularity condition is satisfied, the property of quadratic growth holds and, at least locally, the solutions of the optimization problem and of the associated optimality system coincide. If, in addition the number of inequality constraints is finite, the solution and the Lagrange multiplier can be expanded in Taylor series. If the data are analytic, the solution and the multiplier are analytic functions of the perturbation parameter.     

An analysis of some dual problems in multiobjective optimization (I)

In this work we study the duality for a general multiobjective optimization problem. Considering, first, a scalar problem, different duals using the conjugacy approach are presented. Starting from these scalar duals, we introduce six different multiobjective dual problems to the primal one, one depending on certain vector parameters. The existence of weak and, under certain conditions, strong duality between the primal and the dual problems is shown. Afterwards, some inclusion results for the image sets of the multiobjective dual problems (D ₁), (D _α) and (D_FL ) are derived. Moreover, we verify that the efficiency sets within the image sets of these problems coincide, but the image sets themselves do not.     

Weak Clarke epiderivative in set-valued optimization

A new notion of weak Clarke epiderivative for a set-valued map is introduced using the concept of Clarke tangent cone. The existence, characterization and properties of weak Clarke epiderivative are then studied. Finally optimality criteria are established for a constrained set-valued optimization problem in terms of weak Clarke epiderivative.     

A Hausdorff-type distance,a directional derivative of a set-valued map and applications in set optimization

Abstract

In this paper, we follow Kuroiwa’s set approach in set optimization, which proposes to compare values of a set-valued objective map F with respect to various set order relations. We introduce a Hausdorff-type distance relative to an ordering cone between two sets in a Banach space and use it to define a directional derivative for F. We show that the distance has nice properties regarding set order relations and the directional derivative enjoys most properties of the one of a scalar single-valued function. These properties allow us to derive necessary and/or sufficient conditions for various types of maximizers and minimizers of F.     

Optimality conditions for set-valued maps with set optimization

Problems in set-valued optimization can be solved via set optimization. In this paper optimality conditions are studied for set-valued maps with set optimization. Optimality requirements are established for continuous selections using directional derivatives. Necessary and sufficient conditions for the existence of solutions are shown for set-valued maps under generalized convexity assumptions and with the notion of the contingent derivative.     

Conjugate duality for multiobjective composed optimization problems

Given a multiobjective optimization problem with the components of the objective function as well as the constraint functions being composed convex functions, we introduce, by using the Fenchel-Moreau conjugate of the functions involved, a suitable dual problem. Under a standard constraint qualification and some convexity as well as monotonicity conditions we prove the existence of strong duality. Finally, some particular cases of this problem are presented.      

Nonconvex scalarization in set optimization with set-valued maps

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: Proceedings of the International Conference on Nonlinear Analysis and Convex Analysis, World Scientific, River Edge, NJ, 1999, pp. 221-228], which is based on ordered relations between sets, we characterize this type of solutions by means of nonlinear scalarization. The scalarizing function is a generalization of the Gerstewitz's nonconvex separation function. As applications of our results we give two existence theorems for set-valued optimization problems.     

Super efficiency in vector optimization with nearly convexlike set-valued maps

In this paper, we establish a scalarization theorem and a Lagrange multiplier theorem for super efficiency in vector optimization problem involving nearly convexlike set-valued maps. A dual is proposed and duality results are obtained in terms of super efficient solutions. A new type of saddle point, called super saddle point, of an appropriate set-valued Lagrangian map is introduced and is used to characterize super efficiency.