On the Density of Positive Proper Efficient Points in a Normed Space

In the context of vector optimization and generalizing cones with bounded bases, we introduce and study quasi-Bishop-Phelps cones in a normed space X. A dual concept is also presented for the dual space X*. Given a convex subset A of a normed space X partially ordered by a closed convex cone S with a base, we show that, if A is weakly compact, then positive proper efficient points are sequentially weak dense in the set E(A, S) of efficient points of A; in particular, the connotation weak dense in the above can be replaced by the connotation norm dense if S is a quasi-Bishop-Phelps cone. Dually, for a convex subset of X* partially ordered by the dual cone S ⁺, we establish some density results of positive weak* efficient elements of A in E(A, S ⁺).     

Scalarization of Henig Proper Efficient Points in a Normed Space

In a general normed space equipped with the order induced by a closed convex cone with a base, using a family of continuous monotone Minkowski functionals and a family of continuous norms, we obtain scalar characterizations of Henig proper efficient points of a general set and a bounded set, respectively. Moreover, we give a scalar characterization of a superefficient point of a set in a normed space equipped with the order induced by a closed convex cone with a bounded base.     

Cones Admitting Strictly Positive Functionals and Scalarization of Some Vector Optimization Problems

This paper is concerned with cones admitting strictly positive functionals and scalarization methods in multiobjective optimization. Assuming that the ordering cone admits strictly positive functionals or possesses a base in normed spaces or is a supernormal cone in a Banach space, we give scalar and scalar proper representations for vector optimization problems with convex and naturally quasiconvex data.     

On the existence of efficient points in locally convex spaces

We study the existence of efficient points in a locally convex space ordered by a convex cone. New conditions are imposed on the ordering cone such that for a set which is closed and bounded in the usual sense or with respect to the cone, the set of efficient points is nonempty and the domination property holds.     

Density theorems for generalized Henig proper efficiency

We develop a new, simple technique of proof for density theorems (i.e.,for the sufficient conditions to guarantee that the proper efficient points of a set are dense in the efficient frontier) in an ordered topological vector space. The results are the following: (i) the set of proper efficient points of any compact setQ is dense in the set of efficient points with respect to the original topology of the space whenever the ordering coneK is weakly closed and admits strictly positive functionals; moreover, ifK is not weakly closed, then there exists a compact set for which the density statement fails; (ii) ifQ is weakly compact, then we have only weak density, but ifK has a closed bounded base, then we can assert the density with respect to the original topology, (iii) there exists a similar possibility to assert the strong density for weakly compactQ if additional restrictions are placed onQ instead ofK. These three results are obtained in a unified way as corollaries of the same statement. In this paper, we use the concept of proper efficiency due to Henig. We extend his definition to the setting of a Hausdorff topological vector space.Research of the first author was supported by the Foundation of Fundamental Research of the Republic of Belarus. Authors are grateful to Professor Valentin V. Gorokhovik for suggesting the problem studied in this paper and for numerous fruitful conversations.     

Unified Representation of Proper Efficiency by Means of Dilating Cones

We reduce the definitions of proper efficiency due to Hartley, Henig, Borwein, and Zhuang to a unified form based on the notion of a dilating cone, i.e., an open cone containing the ordering cone. This new form enables us to obtain a comprehensive comparison among these and other kinds of proper efficiency. The most advanced results are obtained for a special class of proper efficiencies corresponding to one-parameter families of uniform dilations. This class is sufficiently wide and includes, for example, the Hartley and Henig proper efficiencies as well as superefficiency.     

Superefficiency in Local Convex Spaces

In the framework of normed spaces, Borwein and Zhuang introduced superefficiency and gave its concise dual form when the underlying decision problem is convex. In this paper, we consider four different generalizations of the Borwein and Zhuang superefficiency in locally convex spaces and give their concise dual forms for convex vector optimization. When the ordering cone has a base, we clarify the relationship between Henig efficiency and the various kinds of superefficiency. Finally, we show that whether the four kinds of superefficiency are equivalent to each other depends on the normability of the underlying locally convex spaces.     

Geometry of cones and an application in the theory of Pareto efficient points

In this article we give a new criterion for the existence of a bounded base for a cone P of a normed space X. Also, if P is closed, we give a partial answer to the problem: is 0 a point of continuity of P if and only if 0 is a denting point of P? The above problems have applications in the theory of Pareto efficient points.     

On Connectedness of Efficient Solution Sets for a Star Cone-Quasiconvex Vector Optimization Problem

In this note,we prove that the efficient solution set for a vector optimization problem with acontinuous,star cone-quasiconvex objective mapping is connected under the assumption that the ordering coneis a D-cone.A D-cone includes any closed convex pointed cones in a normed space which admits strictly positivecontinuous linear functionals.     

ON THE THEOREM OF ARROW-BARANKIN-BLACKWELL FOR WEAKIY COMPACT CONVEX SET

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ON THE THEOREM OF ARROW-BARANKIN-BLACKWELL FOR WEAKLY COMPACT CONVEX SET

This papeer studies the known density theorem of Arrow-Barankin-Blackwell. The following main result is obtained: If X is a Hausdorff locally convex Topological space and C belong to X is a closed convex cone with bounded base, then for every nonempty weakly compact convex subset A, the set of positive proper efficient points of A is dense in the set of efficient points of A.     

Generalization of the Arrow–Barankin–Blackwell Theorem in a Dual Space Setting

In this note, we provide general sufficient conditions under which, if F is a compact [resp. w*-compact] subset of the topological dual Y* of a nonreflexive normed space Y partially ordered by a closed convex pointed cone K, then the set of points in F that can be supported by strictly positive elements in the canonical embedding of Y in Y** is norm dense [resp. w*-dense] in the efficient [maximal] point set of F. This result gives an affirmative answer to the conjecture proposed by Gallagher (Ref. 19), and also generalizes the results stated in Ref. 19 and some space specific results given in Refs. 17, 18, and 11.     

Characterization of properly optimal elements with variable ordering structures

In vector optimization with a variable ordering structure, the partial ordering defined by a convex cone is replaced by a whole family of convex cones, one associated with each element of the space. In recent publications, it was started to develop a comprehensive theory for these vector optimization problems. Thereby, also notions of proper efficiency were generalized to variable ordering structures. In this paper, we study the relation between several types of proper optimality. We give scalarization results based on new functionals defined by elements from the dual cones which allow complete characterizations also in the nonconvex case.     

Some Geometrical Aspects of Efficient Points in Vector Optimization

We study efficient point sets in terms of extreme points, positive support points and strongly positive exposed points. In the case when the ordering cone has a bounded base, we prove that the efficient point set of a weakly compact convex set is contained in the closed convex hull of its strongly positive exposed points, thereby extending the Phelps theorem. We study also the density of positive proper efficient point sets. This research was supported by a Central Research Grant of Hong Kong Polytechnic University, Grant G-T 507. Research of the first author was also supported by the National Natural Science Foundation of P.R. China, Grant 10361008, and the Natural Science Foundation of Yunnan Province, China, Grant 2003A002M. Research of the second author was also supported by the Natural Science Foundation of Chongqing. Research of the third author was supported by a research grant from Australian Research Counsil.     

关于真有效点集的连通性

在局部有界的Ｈａｕｓｄｏｒｆｆ局部凸空间中讨论了集合的真有效点集的连通性问题。证明了当序锥具有基底时,任何非空紧凸集的真有效点集是连通的。     

Solutions of Fuzzy Multiobjective Programming Problems Based on the Concept of Scalarization

Scalarization of fuzzy multiobjective programming problems using the embedding theorem and the concept of convex cone (ordering cone) is proposed in this paper. Since the set of all fuzzy numbers can be embedded into a normed space, this motivation naturally inspires us to invoke the scalarization techniques in vector optimization problems to evaluate the a multiobjective programming problem. Two solution concepts are proposed in this paper by considering different convex cones.     

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.     

Normality and modulability indices. Part I: Convex cones in normed spaces

This paper proposes and compares several ways of measuring the degree of normality of a convex cone contained in a normed space. The dual concept of modulability is also considered. Other notions like solidity and sharpness are also analyzed from a quantitative point of view.     

Globally proper saddle point in ic-cone-convexlike set-valued optimization problems

The existence conditions of globally proper efficient points and a useful property of ic- cone-convexlike set-valued maps are obtained. Under the assumption of the ic-cone-convexlikeness, the optimality conditions for globally proper efficient solutions are established in terms of Lagrange multipliers. The new concept of globally proper saddle-point for an appropriate set-valued Lagrange map is introduced and used to characterize the globally proper efficient solutions. The results which are obtained in this paper are proven under the conditions that the ordering cone need not to have a nonempty interior.     

On farthest points of sets

For a convex closed bounded set in a Banach space, we study the existence and uniqueness problem for a point of this set that is the farthest point from a given point in space. In terms of the existence and uniqueness of the farthest point, as well as the Lipschitzian dependence of this point on a point in space, we obtain necessary and su.cient conditions for the strong convexity of a set in several infinite-dimensional spaces, in particular, in a Hilbert space. A set representable as the intersection of closed balls of a fixed radius is called a strongly convex set. We show that the condition “for each point in space that is sufficiently far from a set, there exists a unique farthest point of the set” is a criterion for the strong convexity of a set in a finite-dimensional normed space, where the norm ball is a strongly convex set and a generating set.