Loose saddle points of set-valued maps in topological vector spaces

We give a new existence theorem for loose saddle point of set-valued map having values in a partially ordered topological vector space which is based on continuity and quasiconvexity- quasiconcavity of its scalarized maps. Moreover, we prove a new saddle point theorem for vector-valued functions in locally convex topological vector spaces under weak condition that is the semicontinuity of two function scalarization.     

Epi and coepi-analysis of one class of vector-valued mappings

This article deals with a new characterization of lower semicontinuity of vector-valued mappings in normed spaces. We study the link between the lower semicontinuity property of vector-valued mappings and the topological properties of their epigraphs and coepigraphs, respectively. We show that if the objective space is partially ordered by a pointed cone with nonempty interior, then coepigraphs are stable with respect to the procedure of their closure and, moreover, the locally semicompact vector-valued mappings with closed coepigraphs are lower semicontinuous. Using these results we propose some regularization schemes for vector-valued functions. In the case when there are no assumptions on the topological interior of the ordering cone, we introduce a new concept of lower semicontinuity for vector-valued mappings, the so-called epi-lower semicontinuity, which is closely related to the closedness of epigraphs of such mappings, and study their main properties. All principal notions and assertions are illustrated by numerous examples.     

Some Deterministic and Random Vector Equilibrium Problems

In this paper, some generalized concepts about semicontinuity and convexity for vector-valued bifunctions are introduced. A new existence theorem for vector equilibrium problems is proved. Further, a random version of this result is considered. Applications to cone saddle points, random vector optimization, random vector variational inequalities, and random vector best approximation problems are finally given.     

Lower Semicontinuous Regularization for Vector-Valued Mappings

The paper is devoted to studying the lower semicontinuity of vector-valued mappings. The main object under consideration is the lower limit. We first introduce a new definition of an adequate concept of lower and upper level sets and establish some of their topological and geometrical properties. A characterization of semicontinuity for vector-valued mappings is thereafter presented. Then, we define a concept of vector lower limit, proving its lower semicontinuity, and furnishing in this way a concept of lower semicontinuous regularization for mappings taking their values in a complete lattice. The results obtained in the present work subsume the standard ones when the target space is finite dimensional. In particular, we recapture the scalar case with a new flexible proof. In addition, extensions of usual operations of lower and upper limits for vector-valued mappings are explored. The main result is finally applied to obtain a continuous D.C. decomposition of continuous D.C. mappings. Dedicated to Alex Rubinov in honor of his 65th birthday     

Existence theorems for cone saddle points of vector-valued functions in infinite-dimensional spaces

For vector-valued functions, cone saddle points are defined, and some existence theorems for them are established in infinite-dimensional spaces. Most of our results rely on Condition 2.1, which has been given by Sterna-Karwat. In particular, it shows that the scalarization of vector-valued functions plays an important role for the condition. Some interesting examples in infinite-dimensional spaces are presented. Moreover, necessary conditions for the existence of cone saddle points are investigated.The author would like to thank the referees for their valuable suggestions on the original draft.     

Existence theorems for saddle points of vector-valued maps

In this paper, we prove some new existence theorems for loose saddle points and for saddle points of set-valued maps or vector-valued functions. These theorems generalize the corresponding results of Tanaka and those of Luc and Vargas via different proofs.The authors would like to thank two referees for their careful reading of the paper and helpful comments.     

Some minimax theorems in vector-valued functions

In this paper, we generalize the well-known notions of minimax, maximin, and saddle point to vector-valued functions. Conditions for a vector-valued function to have a generalized saddle point are given. An example is used to illustrate the generalized concepts of minimax, maximin, and saddle point.     

Symmetric strong vector quasi-equilibrium problems

In this paper, we introduce the symmetric strong vector quasi-equilibrium problem. We then demonstrate that the symmetric strong vector quasi-equilibrium problem is solvable under the suitable assumptions. As an application, we get an existence theorem of the strong saddle points of vector-valued functions. In addition, we give a characterization of vector-valued properly quasi-convex functions. This work was supported by the Natural Science Foundation of China and the Natural Science Foundation of Jiangxi Province, China     

A minimax theorem for vector-valued functions in lexicographic order

In this paper, a minimax theorem and a saddle point theorem are obtained for vector-valued functions in the sense of lexicographic order, respectively. An equivalent relationship between the minimax inequality and the saddle point is established. Some examples are given to illustrate our results.     

Tangency conditions for multivalued mappings

We prove that interiority conditions imply tangency conditions for two multivalued mappings from a topological space into a normed vector space. As a consequence, we obtain the lower semicontinuity of the intersection of two multivalued mappings. An application to the epi-upper semicontinuity of the sum of convex vector-valued mappings is given.