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.      

Characterizations of Solutions for Vector Equilibrium Problems

In this paper, we characterize the solutions of vector equilibrium problems as well as dual vector equilibrium problems. We establish also vector optimization problem formulations of set-valued maps for vector equilibrium problems and dual vector equilibrium problems, which include vector variational inequality problems and vector complementarity problems. The set-valued maps involved in our formulations depend on the data of the vector equilibrium problems, but not on their solution sets. We prove also that the solution sets of our vector optimization problems of set-valued maps contain or coincide with the solution sets of the vector equilibrium problems.     

On gap functions for nonsmooth multiobjective optimization problems

A set-valued gap function, \(\phi \), existing in the literature for smooth and nonsmooth multiobjective optimization problems is dealt with. It is known that \(0\in \phi (x^*)\) is a sufficient condition for efficiency of a feasible solution \(x^*\), while the converse does not hold. In the current work, the converse of this assertion is proved for properly efficient solutions. Afterwards, to avoid the complexities of set-valued maps some new single-valued gap functions, for nonsmooth multiobjective optimization problems with locally Lipschitz data are introduced. Important properties of the new gap functions are established.     

Derivatives of Set-Valued Maps and Gap Functions for Vector Equilibrium Problems

This paper deals with the set-valued gap functions for vector equilibrium problems and investigates their differential properties using Hadamard directional differentials. Also, contingent and adjacent derivatives of a class of set-valued maps are characterized. Moreover, some basic properties of Φ-contingent and Φ-adjacent cones are given in the presence of a nonsmooth kernel function.     

Normal subdifferentials of efficient point multifunctions in parametric vector optimization

The normal subdifferential of a set-valued mapping with values in a partially ordered Banach space has been recently introduced in Bao and Mordukhovich (Control Cyber 36:531–562, 2007), by using the Mordukhovich coderivative of the associated epigraphical multifunction, which has proven to be useful in deriving necessary conditions for super efficient points of vector optimization problems. In this paper, we establish new formulae for computing and/or estimating the normal subdifferential of the efficient point multifunctions of parametric vector optimization problems. These formulae will be presented in a broad class of conventional vgector optimization problems with the presence of geometric, operator, equilibrium, and (finite and infinite) functional constraints.     

On System of Generalized Vector Quasi-equilibrium Problems with Set-valued Maps

In this paper, we introduce four new types of the system of generalized vector quasi-equilibrium problems with set-valued maps which include system of vector quasi-equilibrium problems, system of vector equilibrium problems, system of variational inequality problems, and vector equilibrium problems in the literature as special cases. We prove the existence of solutions for such kinds of system of generalized vector quasi-equilibrium problems. Consequently, we derive some existence results of a solution for the system of vector quasi-equilibrium problems and the generalized Debreu type equilibrium problem for vector-valued functions.     

Refinements of existence results for relaxed quasimonotone equilibrium problems

We consider a general equilibrium problem in a normed vector space setting and we establish sufficient conditions for the existence of solutions in compact and non compact cases. Our approach is based on the concept of upper sign property for bifunctions, which turns out to be a very weak assumption for equilibrium problems. In the framework of variational inequalities, this notion coincides with the upper sign continuity for a set-valued operator introduced by Hadjisavvas. More in general, it allows to strengthen a number of existence results for the class of relaxed $\mu $ -quasimonotone equilibrium problems.     

Scalarization of Set-Valued Optimization Problems with Generalized Cone Subconvexlikeness in Real Ordered Linear Spaces

In real ordered linear spaces, an equivalent characterization of generalized cone subconvexlikeness of set-valued maps is firstly established. Secondly, under the assumption of generalized cone subconvexlikeness of set-valued maps, a scalarization theorem of set-valued optimization problems in the sense of ?-weak efficiency is obtained. Finally, by a scalarization approach, an existence theorem of ?-global properly efficient element of set-valued optimization problems is obtained. The results in this paper generalize and improve some known results in the literature.     

Strongly convex set-valued maps

We introduce the notion of strongly $t$ -convex set-valued maps and present some properties of it. In particular, a Bernstein–Doetsch and Sierpiński-type theorems for strongly midconvex set-valued maps, as well as a Kuhn-type result are obtained. A representation of strongly $t$ -convex set-valued maps in inner product spaces and a characterization of inner product spaces involving this representation is given. Finally, a connection between strongly convex set-valued maps and strongly convex sets is presented.     

一类隐式集值向量均衡问题解的存在性和解集的稳定性

研究一类隐式集值向量均衡问题,它是隐式向量均衡问题、隐式变分不等式问题、隐式相补问题、向量均衡问题和向量变分不等式问题等的推广.利用截口定理,在Hausdorff拓扑线性空间的非紧子集上得出了一些隐式集值向量均衡问题解的存在性结果.并且还讨论了该隐式集值向量均衡问题解集的通有稳定性.这些结果推广和统一了近期的一些相关结果.