Second-Order Optimality Conditions for Set-Valued Vector Equilibrium Problems

In this article, by using the generalized second-order contingent (adjacent) epiderivatives of set-valued maps, we obtain necessary optimality conditions and sufficient optimality conditions for weakly efficient solutions, Henig efficient solutions to the set-valued vector equilibrium problems with constraints. Some results of this article improve the corresponding results in literatures by lessening the assumption of convexity.     

Further on Set-Valued Equilibrium Problems and Applications to Browder Variational Inclusions

In this paper, we introduce some concepts of convexity and semicontinuity for real set-valued mappings similar to those of real single-valued mappings. Then, we obtain different results on the existence of solutions of set-valued equilibrium problems generalizing in a common way several old ones for both single-valued and set-valued equilibrium problems. Applications to Browder variational inclusions, with weakened conditions on the involved set-valued operator, are given.     

Vector Equilibrium Problems. Existence Theorems and Convexity of Solution Set

The natural quasi-concavity of set-valued mappings in an ordered vector space is introduced. Existence theorems for vector equilibrium problems involving set-valued monotone mappings are obtained and the convexity of the solution set is shown.     

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.     

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.      

一个标量泛函的研究及其在集值优化问题中的应用

本文在K条件下,研究了所给标量泛函的连续性和拟凸性,并利用该标量泛函,将集值优化问题转化为均衡问题,进而研究了含约束的集值优化问题弱充分解的存在性和拟集值优化问题强逼近解映射的上半连续性与下半连续性.与最近的文献相比,我们的方法是新的,条件和结论也更具一般性.     

Strong Duality with Proper Efficiency in Multiobjective Optimization Involving Nonconvex Set-Valued Maps

In this paper, we consider some dual problems of a primal multiobjective problem involving nonconvex set-valued maps. For each dual problem, we give conditions under which strong duality between the primal and dual problems holds in the sense that, starting from a Benson properly efficient solution of the primal problem, we can construct a Benson properly efficient solution of the dual problem such that the corresponding objective values of both problems are equal. The notion of generalized convexity of set-valued maps we use in this paper is that of near-subconvexlikeness.     

Uniform convexity and the splitting problem for selections

We continue to investigate cases when the Repovš–Semenov splitting problem for selections has an affirmative solution for continuous set-valued mappings. We consider the situation in infinite-dimensional uniformly convex Banach spaces. We use the notion of Polyak of uniform convexity and modulus of uniform convexity for arbitrary convex sets (not necessary balls). We study general geometric properties of uniformly convex sets. We also obtain an affirmative solution of the splitting problem for selections of certain set-valued mappings with uniformly convex images.     

A New Variant of Ekeland's Variational Principle for Set-Valued Maps

In this article, following the idea used by Göpfert et al . [A. Göpfert, Chr. Tammer and C. Zalinescu (2000). On the vectorial Ekeland's variational principle and minimal points in product spaces. Nonlinear Analysis, Theory, Methods & Applications , 39 , 909-922] to derive an Ekeland's variational principle for vector-valued functions, we derive a new variant of Ekeland's variational principle for set-valued maps. Finally, we apply this variational principle to obtain an approximate necessary optimality condition for a class of set-valued optimization problems.     

近似锥一次类凸集值向量优化问题强有效解的广义鞍点刻画

本文研究了近似锥一次类凸集值向量优化强有效解的广义鞍点表示问题.利用择一定理,得到了近似锥-次类凸集值优化问题强有效解为广义鞍点的充分条件和必要条件.所得结果丰富了集值优化理论,并且拓广了广义鞍点的应用.     

Minimum selections and fixed points of set-valued operators in Banach spaces with some uniform convexity

A new continuity theorem of minimum selection is presented for a continuous set-valued operator from a topological space into a Banach space with some uniform convexity. As applications, some problems concerning minimum right inverses for linear operators and minimum fixed points for condensing set-valued nonlinear operators are discussed. Also, the existence of minimum solutions for an integral inclusion is proved.     

Lagrangian Duality in Set-Valued Optimization

In this paper, we study optimization problems where the objective function and the binding constraints are set-valued maps and the solutions are defined by means of set-relations among all the images sets (Kuroiwa, D. in Takahashi, W., Tanaka, T. (eds.) Nonlinear analysis and convex analysis, pp. 221–228, 1999). We introduce a new dual problem, establish some duality theorems and obtain a Lagrangian multiplier rule of nonlinear type under convexity assumptions. A necessary condition and a sufficient condition for the existence of saddle points are given. The authors thank the two referees for valuable comments and suggestions on early versions of the paper. The research of the first author was partially supported by Ministerio de Educación y Ciencia (Spain) Project MTM2006-02629 and by Junta de Castilla y León (Spain) Project VA027B06.     

On set-valued stochastic integrals in an M-type 2 Banach space

In a separable Banach space, for set-valued martingale, several equivalent conditions based on the measurable selections are discussed, and then, in an M-type 2 Banach space, at first we define single valued stochastic integral by the differential of a real valued Brownian motion, after that extend it to set-valued case. We prove that the set-valued stochastic integral becomes a set-valued submartingale, which is different from single valued case, and obtain the Castaing representation theorem for the set-valued stochastic integral, which is applicable for set-valued stochastic differential equations.     

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.     

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。     

Lagrangian conditions for approximate solutions on nonconvex set-valued optimization problems

The purpose of this paper is to consider the set-valued optimization problem in Asplund spaces without convexity assumption. By a scalarization function introduced by Tammer and Weidner (J Optim Theory Appl 67:297–320, 1990), we obtain the Lagrangian condition for approximate solutions on set-valued optimization problems in terms of the Mordukhovich coderivative.     

Stochastic integral with respect to set-valued square integrable martingales

In this paper, we shall firstly illustrate why we should consider integral of a stochastic process with respect to a set-valued square integrable martingale. Secondly, we shall prove the representation theorem of set-valued square integrable martingale. Thirdly, we shall give the definition of stochastic integral of a stochastic process with respect to a set-valued square integrable martingale and the representation theorem of this kind of integrals. Finally, we shall prove that the stochastic integral is a set-valued sub-martingale.     

An equilibrium version of set-valued Ekeland variational principle and its applications to set-valued vector equilibrium problems

By using Gerstewitz functions, we establish a new equilibrium version of Ekeland variational principle, which improves the related results by weakening both the lower boundedness and the lower semi-continuity of the ob jective bimaps. Applying the new version of Ekeland principle, we obtain some existence theorems on solutions for set-valued vector equilibrium problems, where the most used assumption on compactness of domains is weakened. In the setting of complete metric spaces(Z,d), we present an existence result of solutions for set-valued vector equilibrium problems, which only requires that the domain XZ is countably compact in any Hausdorff topology weaker than that induced by d. When(Z, d) is a Féchet space(i.e., a complete metrizable locally convex space), our existence result only requires that the domain XZ is weakly compact. Furthermore, in the setting of non-compact domains, we deduce several existence theorems on solutions for set-valued vector equilibrium problems,which extend and improve the related known results.     

The Ekeland variational principle for set-valued maps involving coderivatives

In this paper we use the Fréchet, Clarke, and Mordukhovich coderivatives to obtain variants of the Ekeland variational principle for a set-valued map F and establish optimality conditions for set-valued optimization problems. Our technique is based on scalarization with the help of a marginal function associated with F and estimates of subdifferentials of this function in terms of coderivatives of F.