向量优化中改进集的一些拓扑性质

主要研究改进集的一些拓扑运算性质。首先在改进集条件下给出了拓扑向量空间中两个非空集之和的拓扑内部的一些运算性质。进一步,利用改进集获得了 Flores-Baz′an 和 Hern′andez 提出的假定 B 的一个加强形式。此外,给出了一些例子对主要结果进行了解释。     

The fundamental continuity theory of optimization on a compact space. I

Optimizers on a compact feasible region are abstractly specified and, as set-valued mappings, are studied for sufficient conditions yielding them (as well as certain associated maps and certain restrictions of all these) continuous, using function space methods. In particular, the study concerns the continuity of the set of optimal solutions as a function of the three arguments: (i) objective function used, (ii) an incentive (or penalty/reward) function imposed, and (iii) an abstract parameter. An interpretation of the mathematical apparatus is suggested and a brief game-theoretic illustration is given.     

Semicontinuity of the solution set of parametric multivalued vector quasiequilibrium problems

In this paper we establish sufficient conditions for the solution set of parametric multivalued vector quasiequilibrium problems to be semicontinuous. All kinds of semicontinuity are considered: lower semicontinuity, upper semicontinuity, Hausdorff upper semicontinuity and closedness. Moreover, we investigate both the “weak” and “strong” solutions of quasiequilibrium problems.     

An existence theorem for Lagrange problems in Banach spaces

We give an existence theorem for a Lagrange problem in a reflexive separable Banach space. We prove the main theorem by means of the direct method of the calculus of variations introduced by Tonelli.This work has been supported by the Laboratorio per la Matematica Applicata, Consiglio Nazionale delle Ricerche, Genova, Italy.The authors wish to thank Professor J. P. Cecconi who has drawn their attention to these problems.     

向量优化中集合的一些相对代数性质和相对拓扑性质

基于 Flores-Baz′an 等人的思想,提出了假设 B1和假设 B2,证明了集合和的相对代数内部等于相对代数内部的和;集合代数闭包与相对代数内部的和等于和的相对代数内部;集合和的相对拓扑内部等于相对拓扑内部的和;集合拓扑闭包与相对拓扑内部的和等于和的相对拓扑内部,建立了集合代数闭包相等与代数内部相等,拓扑闭包相等与拓扑内部相等之间的一些等价关系。     

Asplund空间中方向上导数和序列法紧性质分析法则

研究了广义微分结构中的集合方向Mordukhovich法锥、集值映射的方向上导数,以及集合和集值映射的方向序列法紧性的分析法则. 基于集合方向Mordukhovich法锥的交集法则,在方向内半紧性假设下,建立了集合的方向Mordukhovich法锥、集值映射的方向上导数的分析法则.此外,借助Asplund乘积空间中集合的方向序列法紧性的交集法则, 在方向内半紧性和相应的规范条件下,建立了集合和集值映射的(部分)方向序列法紧性的加法、逆像、复合等法则.     

带有强吸收项的半线性热方程解的相关性质

研究了一类带有强吸收项的半线性热方程的Cauchy问题,得出了解的存在性和惟一性,并利用比较原理给出了解的支集的瞬间收缩性质和解的有限时间淹没性质及指标p的最优性.     

A note on the growth of topological Sidon sets

We give an estimate for the number of elements in the intersection of topological Sidon sets inR ⁿ with compact convex subsets and deduce a necessary and sufficient conditions for an orbit of a linear transformation ofR ⁿ to be a topological Sidon set.     

A Class of Cone Bounded Quasiconvex Mappings in Topological Vector Spaces

Within the context of cone-ordered topological vector spaces, this paper introduces the concepts of cone bounded point and cone bounded set for vector set. With their aid, a class of new cone quasiconvex mappings in topological vector spaces is defined, and their fundamental properties are presented. The relationships between the cone bounded quasiconvex mapping defined in this paper and cone convex mapping, and other known cone quasiconvex mapping are also discussed.     

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.     

Some aspects of dimension theory for topological groups

We discuss dimension theory in the class of all topological groups. For locally compact topological groups there are many classical results in the literature. Dimension theory for non-locally compact topological groups is mysterious. It is for example unknown whether every connected (hence at least 1-dimensional) Polish group contains a homeomorphic copy of $[0,1]$. And it is unknown whether there is a homogeneous metrizable compact space the homeomorphism group of which is 2-dimensional. Other classical open problems are the following ones. Let $G$ be a topological group with a countable network. Does it follow that $\dim G=\mathrm{ind}G=\mathrm{Ind}G$? The same question if $X$ is a compact coset space. We also do not know whether the inequality $\dim (G\times H)\le \dim G+\dim H$ holds for arbitrary topological groups $G$ and $H$ which are subgroups of $\sigma$-compact topological groups. The aim of this paper is to discuss such and related problems. But we do not attempt to survey the literature.     

Some covering properties in topological and uniform spaces

Recent progress in selection principles theory is discussed and is illustrated mainly by the Hurewicz covering property and its strong version, the Gerlits-Nagy property GN(*). Some results that have not been published elsewhere are given with proofs.     

预不变凸模糊集的一些性质

在下半连续的条件下,给出了一个模糊集是预不变凸模糊集的充分条件,并将模糊凸集上相关性质在模糊不变凸集上作了相应的推广.所给出的条件和证明都是新的.     

On equivalent stability properties in semi-infinite optimization

This paper studies noncompact feasible sets of a semi-infinite optimization problem which are defined by finitely many equality constraints and infinitely many inequality constraints. The main result is the equivalence of the overall validity of the Extended Mangasarian Fromovitz Constraint Qualification with certain (topological) stability conditions. Furthermore, two perturbation theorems being of independent interest are presented.This work was supported by the Deutsche Forschungsgemeinschaft under grant Gu 304/1-2.     

Some properties of the reduced modulus in space

We obtain upper bounds for the conformal modulus of a condenser with uniformly perfect plates and for the reduced modulus of a uniformly perfect set E at . For the reduced moduli of α-uniformly perfect sets we prove the continuity property with respect to the kernel convergence of the complements to these sets in the sense of Carathéodory.     

On the Stability of the Solution Sets of General Multivalued Vector Quasiequilibrium Problems

We give sufficient conditions for the semicontinuity of solution sets of general multivalued vector quasiequilibrium problems. All kinds of semicontinuities are considered: lower semicontinuity, upper semicontinuity, Hausdorff upper semicontinuity, and closedness. Moreover, we investigate the weak, middle, and strong solutions of quasiequilibrium problems. Many examples are provided to give more insights and comparisons with recent existing results.     

Some topological properties of folded Petersen graph

The k-dimensional folded Petersen graph, FP_k, is an iterative Cartesian product on the simple Petersen graph. As an essential component of folded Petersen cube, folded Petersen graph has many important properties. In this paper, we prove that the 3k-wide diameter and 3k-fault diameter of k-dimensional folded Petersen graph is either 2k+1 or 2k+2. Furthermore, we show that FP_k is interval monotone.     

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.