Strict feasibility of pseudomonotone set-valued variational inequalities

In this article, we use degree theory developed in Kien et al. [B.T. Kien, M.-M. Wong, N.C. Wong, and J.C. Yao, Degree theory for generalized variational inequalities and applications, Eur. J. Oper. Res. 193 (2009), pp. 12–22.] to prove a result on the existence of solutions to set-valued variational inequality under a weak coercivity condition, provided that the set-valued mapping is upper semicontinuous with nonempty compact convex values. If the set-valued mapping is pseudomonotone in the sense of Karamardian and upper semicontinuous with nonempty compact convex values, it is shown that the set-valued variational inequality is strictly feasible if and only if its solution set is nonempty and bounded.     

集值映射的一种锥凸性及标量化

在一种集合偏序关系下提出了集值映射的标量锥拟凸概念, 讨论了它与各种锥凸性的关系. 然后对恰当锥拟凸性得到了某种水平集意义下的刻画. 同时建立了集值映射的各种锥凸性通过实值单调增加凸函数表示的标量化复合法则. 最后给出了利用Gerstewitz泛函表示的对集值映射的锥拟凸性的标量化刻画.     

集值映射的一种锥凸性及标量化

在一种集合偏序关系下提出了集值映射的标量锥拟凸概念, 讨论了它与各种锥凸性的关系. 然后对恰当锥拟凸性得到了某种水平集意义下的刻画. 同时建立了集值映射的各种锥凸性通过实值单调增加凸函数表示的标量化复合法则. 最后给出了利用Gerstewitz泛函表示的对集值映射的锥拟凸性的标量化刻画.     

Scalar representation and conjugation of set-valued functions

We consider functions with values in the power set of a pre-ordered, separated locally convex space with closed convex images. To each such function, a family of scalarizations is given which completely characterizes the original function. A concept of a Legendre–Fenchel conjugate for set-valued functions is introduced and identified with the conjugates of the scalarizations. To the set-valued conjugate, a full calculus is provided, including a biconjugation theorem, a chain rule and weak and strong duality results of the Fenchel–Rockafellar type.     

Radon-Nikodym theorems for set-valued measures

Set-valued measures whose values are subsets of a Banach space are studied. Some basic properties of these set-valued measures are given. Radon-Nikodym theorems for set-valued measures are established, which assert that under suitable assumptions a set-valued measure is equal (in closures) to the indefinite integral of a set-valued function with respect to a positive measure. Set-valued measures with compact convex values are particularly considered.     

On single-valuedness of convex set-valued maps

It is proved that ifX andY are linear spaces andF :X p(Y) is a set-valued map with convex graph such thatF(x) Ø for allx X andF(x ₀) is a singleton for somex ₀, thenF is single-valued and affine. Applications to metric projections and to adjoints of set-valued maps are given.Supported by NSF Grant DMS-9100228.The main result of this paper has been obtained while the second author was visiting the Pennsylvania State University in the framework of the exchange agreement between the Romanian Academy and the National Academy of Sciences of the U.S.A.     

On super efficiency in set-valued optimization

The set-valued optimization problem with constraints is considered in the sense of super efficiency in locally convex linear topological spaces. Under the assumption of iccone-convexlikeness, by applying the seperation theorem, Kuhn-Tucker＇s, Lagrange＇s and saddle points optimality conditions, the necessary conditions are obtained for the set-valued optimization problem to attain its super efficient solutions. Also, the sufficient conditions for Kuhn-Tucker＇s, Lagrange＇s and saddle points optimality conditions are derived.     

Equilibria of set-valued maps on nonconvex domains

We present new theorems on the existence of equilibria (or zeros) of convex as well as nonconvex set-valued maps defined on compact neighborhood retracts of normed spaces. The maps are subject to tangency conditions expressed in terms of new concepts of normal and tangent cones to such sets. Among other things, we show that if is a compact neighborhood retract with nontrivial Euler characteristic in a Banach space , and is an upper hemicontinuous set-valued map with nonempty closed convex values satisfying the tangency condition

then there exists such that Here, denotes a new concept of retraction tangent cone to at suited for compact neighborhood retracts. When is locally convex at coincides with the usual tangent cone of convex analysis. Special attention is given to neighborhood retracts having ``lipschitzian behavior', called retracts below. This class of sets is very broad; it contains compact homeomorphically convex subsets of Banach spaces, epi-Lipschitz subsets of Banach spaces, as well as proximate retracts. Our results thus generalize classical theorems for convex domains, as well as recent results for nonconvex sets.

     

Optimality conditions of the set-valued optimization problem with generalized cone convex set-valued maps characterized by contingent epiderivative

In this paper, firstly, a new notion of generalized cone convex set-valued map is introduced in real normed spaces. Secondly, a property of the generalized cone convex set-valued map involving the contingent epiderivative is obtained. Finally, as the applications of this property, we use the contingent epiderivative to establish optimality conditions of the set-valued optimization problem with generalized cone convex set-valued maps in the sense of Henig proper efficiency. The results obtained in this paper generalize and improve some known results in the literature.     

Some useful set-valued maps in set optimization

In this article, we introduce several classes of set-valued maps which can be useful in set optimization due to their applications. Exactly, we present some set-valued maps defined by scalar and vector functions and study their properties such as continuity and convexity among others. In addition, we compute their asymptotic maps which can be employed to establish coercivity and existence results in the framework of set optimization problems. Finally, we expose some possible directions for further research.     

Connectedness of super efficient solution sets for set-valued maps in Banach spaces

In this paper, we study the connectedness of the super efficient solution sets in convex vector optimization for set-valued maps in Banach spaces.     

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.     

Properties of the metric projection on weakly vial-convex sets and parametrization of set-valued mappings with weakly convex images

We continue studying the class of weakly convex sets (in the sense of Vial). For points in a sufficiently small neighborhood of a closed weakly convex subset in Hubert space, we prove that the metric projection on this set exists and is unique. In other words, we show that the closed weakly convex sets have a Chebyshev layer. We prove that the metric projection of a point on a weakly convex set satisfies the Lipschitz condition with respect to a point and the Hölder condition with exponent 1/2 with respect to a set. We develop a method for constructing a continuous parametrization of a set-valued mapping with weakly convex images. We obtain an explicit estimate for the modulus of continuity of the parametrizing function.     

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.     

Equilibria for set-valued maps on nonsmooth domains

A set-valued map defined on a compact lipschitzian retract of a normed space with nontrivial Euler characteristic and satisfying (i) a strong graph approximation property and (ii) a tangency condition expressed in terms of Clarke’s tangent cone, admits an equilibrium. This result extends in a simple way known solvability theorems to a large class of nonconvex set-valued maps defined on nonsmooth domains. Dedicated to Professor Felix Browder     

A Leray-Schauder type theorem for approximable maps: a simple proof

We present a simple and direct proof for a Leray-Schauder type alternative for a large class of condensing or compact set-valued maps containing convex as well as nonconvex maps.

     

Super efficient solutions for set-valued maps

Some properties for K-preinvex set-valued maps in terms of normal subdifferential are obtained. Furthermore, some sufficient conditions for existence of super minimal points and necessary optimality conditions for a general kind of super efficiency are established.     

Characterizations of the nonemptiness and compactness for solution sets of convex set-valued optimization problems

In this paper, we first derive several characterizations of the nonemptiness and compactness for the solution set of a convex scalar set-valued optimization problem (with or without cone constraints) in which the decision space is finite-dimensional. The characterizations are expressed in terms of the coercivity of some scalar set-valued maps and the well-posedness of the set-valued optimization problem, respectively. Then we investigate characterizations of the nonemptiness and compactness for the weakly efficient solution set of a convex vector set-valued optimization problem (with or without cone constraints) in which the objective space is a normed space ordered by a nontrivial, closed and convex cone with nonempty interior and the decision space is finite-dimensional. We establish that the nonemptiness and compactness for the weakly efficient solution set of a convex vector set-valued optimization problem (with or without cone constraints) can be exactly characterized as those of a family of linearly scalarized convex set-valued optimization problems and the well-posedness of the original problem.     

Multi-segmental representations and approximation of set-valued functions with 1D images

In this work univariate set-valued functions (SVFs, multifunctions) with 1D compact sets as images are considered. For such a continuous SFV of bounded variation (CBV multifunction), we show that the boundaries of its graph are continuous, and inherit the continuity properties of the SVF. Based on these results we introduce a special class of representations of CBV multifunctions with a finite number of ‘holes’ in their graphs. Each such representation is a finite union of SVFs with compact convex images having boundaries with continuity properties as those of the represented SVF. With the help of these representations, positive linear operators are adapted to SVFs. For specific positive approximation operators error estimates are obtained in terms of the continuity properties of the approximated multifunction.