On the density of proper efficient points

In this paper, our aim is to discuss the density of proper efficient points. As an interesting application of the results in this paper, we want to prove a density theorem of Arrow, Barankin, and Blackwell.

     

A generalization of a theorem of Arrow,Barankin and Blackwell to a nonconvex case

The paper presents a generalization of a known density theorem of Arrow, Barankin, and Blackwell for properly efficient points defined as support points of sets with respect to monotonically increasing sublinear functions. This result is shown to hold for nonconvex sets of a partially ordered reflexive Banach space.     

广义的Arrow-Barankin-Blackwell定理

本文首次引进了凸锥广义基的概念,然后将局部凸空间的Ａｒｒｏｗ－Ｂａｒａｎｋｉｎ－Ｂｌａｃｋｗｅｌｌ定理推广了到序凸锥为非点式锥的情况。     

Connectedness of efficient points in convex and convex transformable vector optimization

Given a convex vector optimization problem with respect to a closed ordering cone, we show the connectedness of the efficient and properly efficient sets. The Arrow–Barankin–Blackwell theorem is generalized to nonconvex vector optimization problems, and the connectedness results are extended to convex transformable vector optimization problems. In particular, we show the connectedness of the efficient set if the target function f is continuously transformable, and of the properly efficient set if f is differentiably transformable. Moreover, we show the connectedness of the efficient and properly efficient sets for quadratic quasiconvex multicriteria optimization problems.     

The structure of admissible points with respect to cone dominance

We study the set of admissible (Pareto-optimal) points of a closed, convex setX when preferences are described by a convex, but not necessarily closed, cone. Assuming that the preference cone is strictly supported and making mild assumptions about the recession directions ofX, we extend a representation theorem of Arrow, Barankin, and Blackwell by showing that all admissible points are either limit points of certainstrictly admissible alternatives or translations of such limit points by rays in the closure of the preference cone. We also show that the set of strictly admissible points is connected, as is the full set of admissible points.Relaxing the convexity assumption imposed uponX, we also consider local properties of admissible points in terms of Kuhn-Tucker type characterizations. We specify necessary and sufficient conditions for an element ofX to be a Kuhn-Tucker point, conditions which, in addition, provide local characterizations of strictly admissible points.Several results from this paper were presented in less general form at the National ORSA/TIMS Meeting, Chicago, Illinois, 1975.This research was supported, in part, by the United States Army Research Office (Durham), Grant No. DAAG-29-76-C-0064, and by the Office of Naval Research, Grant No. N00014-67-A-0244-0076. The research of the second author was partially conducted at the Center for Operations Research and Econometrics (CORE), Université Catholique de Louvain, Heverlee, Belgium.The authors are indebted to A. Assad for several helpful discussions and to A. Weiczorek for his careful reading of an earlier version of this paper.     

Maximal Points of Convex Sets in Locally Convex Topological Vector Spaces: Generalization of the Arrow–Barankin–Blackwell Theorem

In 1953, Arrow, Barankin, and Blackwell proved that, if C is a nonempty compact convex set in Rⁿ with its standard ordering, then the set of points in C maximizing strictly positive linear functionals is dense in the set of maximal points of C. In this paper, we present a generalization of this result. We show that that, if C is a compact convex set in a locally convex topological space X and if K is an ordering cone on X such that the quasi-interiors of K and the dual cone K* are nonempty, then the set of points in C maximizing strictly positive linear functionals is dense in the set of maximal points of C. For example, our work shows that, under the appropriate conditions, the density results hold in the spaces Rⁿ, L^p(, ), 1p, l^p, 1p, and C (), a compact Hausdorff space, when they are partially ordered with their natural ordering cones.     

The Arrow-Barankin-Blackwell theorem in a dual space setting

In 1953 Arrow, Barankin, and Blackwell proved that, ifR ⁿ is equipped with its natural ordering and ifF is a closed convex subset ofR ⁿ , then the set of points inF that can be supported by strictly positive linear functionals is dense in the set of all efficient (maximal) points ofF. Many generalizations of this density result to infinite-dimensional settings have been given. In this note, we consider the particular setting where the setF is contained in the topological dualY ^* of a partially ordered, nonreflexive normed spaceY, and the support functionals are restricted to be either nonnegative or strictly positive elements in the canonical embedding ofY inY ^*. Three alternative density results are obtained, two of which generalize a space-specific result due to Majumdar for the dual system (Y,Y ^*)=(L ¹,L ).This research was supported in part by funds provided by the Provident Chair of Excellence in Applied Mathematics at the University of Tennessee, Chattanooga, Tennessee.     

Pareto Solutions of Polyhedral-valued Vector Optimization Problems in Banach Spaces

In general Banach spaces, we consider a vector optimization problem (SVOP) in which the objective is a set-valued mapping whose graph is the union of finitely many polyhedra. We establish some results on structure and connectedness of the weak Pareto solution set, Pareto solution set, weak Pareto optimal value set and Pareto optimal value set of (SVOP). In particular, we improve and generalize Arrow, Barankin and Blackwell’s classical results on linear vector optimization problems in Euclidean spaces.     

Blackwell’s theorem for fuzzy variables

Recently, Zhao et al. (Euro J Oper Res 169:189–201, 2006) discussed a random fuzzy renewal process based on the random fuzzy theory and established Blackwell’s theorem in random fuzzy sense. They obtained Blackwell’s theorem for fuzzy variables by degenerating the process. However, this result is invalid. We provide some counterexamples and offer a corrected version of fuzzy Blackwell’s theorem.     

Existence of Efficient Points in Vector Optimization and Generalized Bishop–Phelps Theorem

In a set without linear structure equipped with a preorder, we give a general existence result for efficient points. In a topological vector space equipped with a partial order induced by a closed convex cone with a bounded base, we prove another kind of existence result for efficient points; this result does not depend on the Zorn lemma. As applications, we study a solution problem in vector optimization and generalize the Bishop–Phelps theorem to a topological vector space setting by showing that the B-support points of any sequentially complete closed subset A of a topological vector space E is dense in A, where B is any bounded convex subset of E.     

有效点、弱有效点和真有效点的锥刻画

利用集合在某点的相依切锥、法向锥和可行方向锥等研究向量优化问题的有效点、 弱有效点和真有效点的特征,对局部有效点、局部弱有效点和局部真有效点与集合的各 锥之间的关系作了刻画.     

Cartan–Slodkowski spectra, splitting elements and noncommutative spectral mapping theorems

In this paper, we prove Slodkowski version of the infinite-dimensional spectral mapping theorem and Cartan–Slodkowski version of the finite-dimensional spectral mapping theorem for nilpotent operator Lie subalgebras with respect to the various noncommutative functional calculi.     

Generalization of the Rödseth–Gupta Theorem on Binary Partitions

In this article, we give a new proof of the Rödseth–Gupta theorem on binary partitions and give one possible generalization of this theorem.     

A Tauberian theorem for power series

In this paper we prove a complex Tauberian theorem for functions analytic in the open unit disc.The method used is based on Cauchy's integral formula for the Taylor coefficients. We show how a Blackwell type renewal theorem is an easy consequence of our main result.     

A Borsuk–Ulam Theorem for Maps from a Sphere to a Generalized Manifold

H. J. Munkholm obtained a generalization for topological manifolds of the famous Borsuk–Ulam type theorem proved by Conner and Floyd. The purpose of this paper is to prove a version of Conner and Floyd's theorem for generalized manifolds.     

Renewal theory for extremal Markov sequences of Kendall type

The paper deals with renewal theory for a class of extremal Markov sequences connected with the Kendall convolution. We consider here some particular cases of the Wold processes associated with generalized convolutions. We prove an analogue of the Fredholm theorem for all regular generalized convolutions algebras. Using regularly varying functions we prove a Blackwell theorem and a limit theorem for renewal processes defined by Kendall random walks.Our results set new research hypotheses for other generalized convolution algebras to investigate renewal processes constructed by Markov processes with respect to generalized convolutions.     

Blackwell type theorem for general T-related and identically distributed fuzzy variables

In this paper, we define a fuzzy variable \(N(t+c) \circleddash N(t)\) using \(\alpha -\)cut representation and consider a necessary and sufficient condition of the Blackwell type theorem based on the Hausdorff metric for general T-related and identically distributed fuzzy variables. We also take into consideration the derived fuzzy Blackwell’s Theorem based on the expected value of fuzzy variables.     

Hyperbolic Manifolds, Harmonic Forms, and Seiberg–Witten Invariants

New estimates are derived concerning the behavior of self-dual harmonic 2-forms on a compact Riemannian 4-manifold with nontrivial Seiberg–Witten invariants. Applications include a vanishing theorem for certain Seiberg–Witten invariants on compact 4-manifolds of constant negative sectional curvature.     

On Classes of Generalized Convex Functions, Gordan–Farkas Type Theorems, and Lagrangian Duality

In this paper, we introduce several classes of generalized convex functions already discussed in the literature and show the relation between these classes. Moreover, a Gordan–Farkas type theorem is proved for all these classes and it is shown how these theorems can be used to verify strong Lagrangian duality results in finite-dimensional optimization.     

Invex Functions and Generalized Convexity in Multiobjective Programming

Martin (Ref. 1) studied the optimality conditions of invex functions for scalar programming problems. In this work, we generalize his results making them applicable to vectorial optimization problems. We prove that the equivalence between minima and stationary points or Kuhn–Tucker points (depending on the case) remains true if we optimize several objective functions instead of one objective function. To this end, we define accurately stationary points and Kuhn–Tucker optimality conditions for multiobjective programming problems. We see that the Martin results cannot be improved in mathematical programming, because the new types of generalized convexity that have appeared over the last few years do not yield any new optimality conditions for mathematical programming problems.