Minimax theorems for set-valued maps without continuity assumptions

We introduce several classes of set-valued maps with new generalized convexity properties. We also obtain minimax theorems for set-valued maps which satisfy these convexity assumptions and which are not continuous. Our method consists of the use of a fixed point theorem for weakly naturally quasiconcave set-valued maps, defined on a simplex in a topological vector space, or of a constant selection of quasiconvex set-valued maps.     

Selezioni continue o misurabili

We give a characterization of set-valued mappings from a topological (measure) space into the class of real non-empty intervals admitting a continuous (measurable) selection. As an application we obtain a characterization of set-valued functions defined on IR ⁿ admitting an approximately continuous or an approximately continuous and almost everywhere continuous selection.     

Set-valued mapping monotonicity as characterization of D.C functions

In this paper, using a result of Kung-Ching Chang [6], we give a characterization of locally Lipschitz functions which are differences of convex functions defined on a Banach space (not necessarily Asplund) in terms on maximal cyclically monotone set-valued mappings. A subdifferential integration of locally D.C functions. is also given.     

集值映射的Epi-导数及其应用

在本文里，集值映射的Epi-导数被引入，它可以认作是实值Lipschitz函数的Clarke-广义方向导数的推广，同时它的一些性质也被研究．进一步地，利用这个Epi-导数集值映射的次微分被定义并研究它的性质．作为其应用，我们给出了集值优化问题的一些(必要或充分)最优性条件．     

Some properties of set-valued stochastic integrals

The present paper is devoted to properties of set-valued stochastic integrals defined as some special type of set-valued random variables. In particular, it is shown that if the probability base is separable or probability measure is nonatomic then defined set-valued stochastic integrals can be represented by a sequence of Itô?s integrals of nonanticipative selectors of integrated set-valued processes. Immediately from Michael?s continuous selection theorem it follows that the indefinite set-valued stochastic integrals possess some continuous selections. The problem of integrably boundedness of set-valued stochastic integrals is considered. Some remarks dealing with stochastic differential inclusions are also given.     

集值信息系统在相容关系下的属性约简

借助于属性集值的相似程度在集值信息系统上定义了一种新的相客关系,给出了这种相客关系下集值信息系统的属性约简与判定,得到了集值信息系统属性约简的具体探作方法,并讨论了相似水平对集值信息系统的属性约简的影响.     

Martingale representation theorem for set-valued martingales

The present paper contains a martingale representation theorem for set-valued martingales defined on a filtered probability space with a filtration generated by a Brownian motion. It is proved that such type martingales can be defined by some generalized set-valued stochastic integrals with respect to a given Brownian motion. The main result of the paper is preceded by short part devoted to the definition and some properties of generalized set-valued stochastic integrals.     

W-G-F-KKM mapping, intersection theorems and minimax inequalities in FC-space

By using basic KKM theorem, a new matching theorem and some minimax inequalities for set-valued mappings defined on the FC-spaces are proved under very weak assumptions. These results generalized many known results from the recent literature.     

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.

     

A Projection-Type Method for Set Valued Variational Inequality Problems on Hadamard Manifolds

In this paper, we develop a projection-type algorithm for set-valued variational inequalities on Hadamard manifolds. The proposed method is well defined whether the solution set of the problem is non-empty or not. Under pseudomonotonicity assumptions on the underlying vector field, our method is convergent to a solution of the given set-valued variational inequality. The results presented in this paper generalize and improve some known results introduced by Tang et al. (Optimization 64(5):1081–1096, 2015).     

Conic Set-Valued Maps in Vector Optimization

The paper deals with the properties of a conic set-valued function defined on the set of all ideal points of vector programming problems. The results here about the continuity and derivability of this conic set-valued map, can be used to get information about the sensitivity of the problem and the stability of the order associated to every ideal point. Furthermore, it is proved that certain contingent cones are determined by the ideal conic set-valued map.      

DEMICONTINUITY,GENERALIZED CONVEXITY AND LOOSE SADDLE POINTS OF SET-VALUED MAPS

Abstract

The aim of this paper is to establish the existence of loose saddle points for a set-valued map which is defined in a topological vector space and possesses the demicontinuity and generalized convexity. Our results strengthen several previously obtained results on the existence of saddle and loose saddle points for single- or set-valued maps.     

基于集值映射的形式概念分析

Formal concept analysis (FCA) is a discipline that studied the hierarchical structares induced by a binary relation between a pair of sets,and applies in data analysis,information retrieval,knowledge discovery,etc.In this paper,it is shown that a formal context T is equivalent to a set-valued mapping S :G →P(M),and formal concepts could be defined in the set-valued mapping S.It is known that the topology and set-valued mapping are linked.Hence,the advantage of this paper is that the conclusion make us to construct formal concept lattice based on the topology.     

Absolute fixed point sets for multi-valued maps

The notion of a multi-valued absolute fixed point set (MAFS) will be defined and characterized in the setting of set-valued maps with images containing multiple components.

     

Semigroups of Jensen set-valued functions

A set-valued function F defined on a convex cone S in a real vector space X into the set n(Y) of all non-empty subsets of a real vector space Y is said to be the Jensen function iff $${1\over 2}\lbrack F(x)+F(x)\rbrack=F\lbrack{1\over 2}(x+y)\rbrack$$ for all x, y ∈ S. This note deals with iteration semigroups of Jensen set-valued functions on S.     

Higher-Order Optimality Conditions for Set-Valued Optimization

This paper deals with higher-order optimality conditions of set-valued optimization problems. By virtue of the higher-order derivatives introduced in (Aubin and Frankowska, Set-Valued Analysis, Birkhäuser, Boston, [1990]) higher-order necessary and sufficient optimality conditions are obtained for a set-valued optimization problem whose constraint condition is determined by a fixed set. Higher-order Fritz John type necessary and sufficient optimality conditions are also obtained for a set-valued optimization problem whose constraint condition is determined by a set-valued map.     

Stochastic morphological evolution equations

The inadequacy of locally defined set-valued differential equations to describe the evolution of shapes and morphological forms in biology, which are usually neither convex or nondecreasing, was recognised by J.-P. Aubin, who introduced morphological evolution equations, which are essentially nonlocally defined set-valued differential equations with the inclusion vector field also depending on the entire reachable set. This concept is extended here to the stochastic setting of set-valued Itô evolution equations in Hilbert spaces. Due to the nonanticipative nature of Itô calculus, the evolving reachable sets are nonanticipative nonempty closed random sets. The existence of solutions and their dependence on initial data are established. The latter requires the introduction of a time-oriented semi-metric in time-space variables. As a consequence the stochastic morphological evolution equations generate a deterministic nonautonomous dynamical system formulated as a two-parameter semigroup with the complication that the random subsets take values in different spaces at different time instances due to the nonanticipativity requirement. It is also shown how nucleation processes can be handled in this conceptual framework.     

Some fixed-point theorems and minimax inequalities in FC-space

By applying the existence theorem of maximal elements, some new collectively fixed-point theorems for a family of set-valued mappings defined on the product space of noncompact FC-space are proved and some new theorems about minimax inequality involving two functions are given to show the relations of fixed-point theorem and minimax inequality in FC-spaces. These results improve and generalize many important results in the recent literature.     

Subdivision Schemes of Sets and the Approximation of Set-Valued Functions in the Symmetric Difference Metric

In this work we construct subdivision schemes refining general subsets of ? ⁿ and study their applications to the approximation of set-valued functions. Differently from previous works on set-valued approximation, our methods are developed and analyzed in the metric space of Lebesgue measurable sets endowed with the symmetric difference metric. The construction of the set-valued subdivision schemes is based on a new weighted average of two sets, which is defined for positive weights (corresponding to interpolation) and also when one weight is negative (corresponding to extrapolation). Using the new average with positive weights, we adapt to sets spline subdivision schemes computed by the Lane–Riesenfeld algorithm, which requires only averages of pairs of numbers. The averages of numbers are then replaced by the new averages of pairs of sets. Among other features of the resulting set-valued subdivision schemes, we prove their monotonicity preservation property. Using the new weighted average of sets with both positive and negative weights, we adapt to sets the 4-point interpolatory subdivision scheme. Finally, we discuss the extension of the results obtained in metric spaces of sets, to general metric spaces endowed with an averaging operation satisfying certain properties.     

二层多目标最优化问题解集的连通性

研究了二层多目标最优化模型(BLMOP)解集的连通性问题,其中(BLMOP)的上层集值目标函数由下层问题的有效点确定.把(BLMOP)看作成单层的集值函数优化问题,借助集值函数优化问题各种有效解集的连通性的结论,得到了(BLMOP)相应的有效解集连通性的结论.