Proper Efficient Points Research Articles

Existence and density theorems of Henig global proper efficient points

In this work, we provide some novel results that establish both the existence of Henig global proper efficient points and their density in the efficient set for vector optimization problems in arbitrary normed spaces. Our results do not require the assumption of convexity, and in certain cases, can be applied to unbounded sets. However, it is important to note that a weak compactness condition on the set (or on a section of it) and a separation property between the order cone and its conical neighbourhoods remains necessary. The weak compactness condition ensures that certain convergence properties hold. The separation property enables the interpolation of a family of Bishop-Phelps cones between the order cone and each of its conic neighbourhoods. This interpolation, combined with the proper handling of two distinct types of conic neighbourhoods, plays a crucial role in the proofs of our results, which include as a particular case other results that have already been established under more restrictive conditions.

Sublinear scalarizations for proper and approximate proper efficient points in nonconvex vector optimization

We show that under a separation property, a {{{mathcal {Q}}}}-minimal point in a normed space is the minimum of a given sublinear function. This fact provides sufficient conditions, via scalarization, for nine types of proper efficient points; establishing a characterization in the particular case of Benson proper efficient points. We also obtain necessary and sufficient conditions in terms of scalarization for approximate Benson and Henig proper efficient points. The separation property we handle is a variation of another known property and our scalarization results do not require convexity or boundedness assumptions.

Pareto efficiency without topology

Given a vector optimization problem over spaces endowed with a topological linear structure, existence results for optima (efficient points) are known. Relying only on the linear structure, the set of properly efficient points from a convex set is proved to be nonempty and the sets of Proper efficient points and Pareto efficient points coincide, provided that the set of internal points picked from the corresponding cone is nonempty. This result is appealing since the scalarization of the vector optimization problem is valid without topological requirements. A As an important consequence, we provide the Second Welfare Theorem in vector lattices and especially in Lebesgue spaces holds without topology.

Pareto set reduction based on an axiomatic approach with application of some metrics

A multicriteria choice problem involving a decision-maker’s binary preference relation is considered. Several two-stage methods are proposed for its solution. First, the Pareto set is reduced by applying an axiomatic approach. Then the problem is scalarized on the resulting set by using the Chebyshev or Euclidean metric. The methods proposed are substantiated with the help of well-known and new techniques for characterizing weakly efficient and proper efficient points. Illustrative examples are given.

Global proper efficiency and vector optimization with cone-arcwise connected set-valued maps

This paper deals withthe characteristics of global proper efficient points and theoptimality conditions of vector optimization problems involvinggeneralized convex set-valued maps. Several equivalent properties ofglobal proper efficient points are proposed. Utilizing cone-directedcontingent derivative, it presents the unified necessary andsufficient optimality conditions for global proper efficient elementin vector optimization problem with cone-arcwise connectedset-valued mapping.

The Ekeland variational principle for Henig proper minimizers and super minimizers

In this paper we consider, for the first time, approximate Henig proper minimizers and approximate super minimizers of a set-valued map F with values in a partially ordered vector space and formulate two versions of the Ekeland variational principle for these points involving coderivatives in the sense of Ioffe, Clarke and Mordukhovich. As applications we obtain sufficient conditions for F to have a Henig proper minimizer or a super minimizer under the Palais–Smale type conditions. The techniques are essentially based on the characterizations of Henig proper efficient points and super efficient points by mean of the Henig dilating cones and the Hiriart-Urruty signed distance function.

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 iccone-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.

Henig proper efficient points and generalized Henig proper efficient points

Applying the theory of locally convex spaces to vector optimization, we investigate the relationship between Henig proper efficient points and generalized Henig proper efficient points. In particular, we obtain a sufficient and necessary condition for generalized Henig proper efficient points to be Henig proper efficient points. From this, we derive several convenient criteria for judging Henig proper efficient points.

Dual Characterization and Scalarization for Benson Proper Efficiency

By using dual cones and their properties, we establish a fundamental dual characterization and scalarization for Benson proper efficient points without any additional assumption on the ordering cone. From this, we obtain several scalarization theorems and Lagrange multiplier theorems for Benson proper minimizers of optimization problems with nearly cone-subconvexlike set-valued maps. The related known results are improved, and some new criteria for checking Benson proper efficiency are deduced.

Some Characterizations of Ideal Points in Vector Optimization Problems

Ideal points and efficient elements play an important role in the investigation of multiobjective optimization problems see, e.g., 1–9 and references therein . Recently, some relations between ideal points and efficient elements have been studied by many authors see 3, 4 . In 3 , the authors derived some sufficient conditions for the existence of ideal points in normed vector spaces. The algebra closure and vector closure were investigated in 1, 2 , which are weaker than topological closure. In this paper, we present some relations between proper ideal points and weakly, positive proper, general positive efficient points in real linear spaces. We also derive some sufficient conditions for the existence of proper ideal points and positive proper efficient points. Let X be a real linear space and A a nonempty subset of X. A is said to be a cone if λA ⊂ A for all λ > 0. A is called a convex cone if A is a cone and A A ⊂ A. A is called a pointed cone if A is a cone and A ∩ −A {0}. The algebraic interior and relative algebraic interior ofA ⊂ X are defined by, respectively,

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+).

Existence and Density Results for Proper Efficiency in Cone Compact Sets

Existence and density results are established for positive proper efficient points, Henig proper efficient points, and superefficient points in cone compact sets.

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.

Some More Density Results for Proper Efficiencies

In this paper we give sufficient conditions for a set and/or a cone ensuring the density of Henig's proper efficient points and positive proper efficient points in the set of efficient points in Hausdorff locally convex spaces. These conditions are based on the concept of the point of continuity an d generalize many existing results.

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.

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. In [1], Luc introduced a new concept of the proper efficient point for a set. Using some results of recession cone, Luc established efficiency conditions, especially proper efficiency and domination properties ( [1, 2]). The present paper is devoted to the study of the density of proper efficient points. In detail, the set of proper efficient points for a set is dense in the set of efficient points. As an interesting application of the results in this paper, we prove a density theorem of Arrow, Barankin, and Blackwell ( [3,4]). First let us recall some notations: Throughout the paper, E is a separated locally convex topological linear space and E* its topological dual. U(O) denotes the family of balanced open convex neighbourhoods of the origin in E. For A c E, cone(A), cl(A), and int(A) denote the generated cone, the closure, and the interior of A, respectively. Let C C E be a convex cone, and let A be a nonempty subset of E. We say that x E A is an efficient point of A with respect to C if there exists y E A, such that y E x-C; then y E x + C. Equivalently, (x-C) n A c x + C. If the C is pointed (that is, C n (-C) =' o}), then x E A is an efficient point iff (x C) n A= {x}. We denote by E(A, C) the set of all efficient points of A (with respect to C). We say that x E A is a proper efficient point of A with respect to C if there exists a closed convex cone K =/ E such that C\{O} c int(K) and x E(A, K). The set of proper efficient points of A is denoted by Prop E(A, C). It is obvious that the set of proper efficient points of A is contained in the set of efficient points, Prop E(A, C) C E(A, C), but the converse is not generally true. If C is a convex cone, the convex set B c C is said to be a base of C if O0 cl(B) and C=cone(B)=U{tB:t>0}={tb:t>ObB}. A cone with base must be pointed. Received by the editors December 14, 1993 and, in revised form, October 3, 1994. 1991 Mathematics Subject Classification. Primary 90C31.

Density Results for Proper Efficiencies

Several density results are established for different notions of proper efficiency in vector optimization without the requirement of ordering cones being boundedly based. For a compact set, the set of Henig proper efficient points is shown to be dense in the set of all efficient points. Density theorems for Borwein's proper efficiency and for positive scalarizable efficiency follow immediately with appropriate assumptions.

Bases of convex cones and Borwein's proper efficiency

In this note, we establish some interesting relationships between the existence of Borwein's proper efficient points and the existence of bases for convex ordering cones in normed linear spaces. We show that, if the closed unit ball in a smooth normed space ordered by a convex cone possesses a proper efficient point in the sense of Borwein, then the ordering cone is based. In particular, a convex ordering cone in a reflexive space is based if the closed unit ball possesses a proper efficient point. Conversely, we show that, in any ordered normed space, if the ordering cone has a base, then every weakly compact set possesses a proper efficient point.

Positive Proper Efficient Points and Related Cone Results in Vector Optimization Theory

Positive proper efficient points are defined as solutions of appropriate linear scalar optimization problems. A geometric characterization of positive proper efficient points is given as well as conditions under which the set of positive proper efficient points is dense in the set of all efficient points. It is shown that these results are applicable in the normed vector lattices $C[a,b]$, $l^p $, and $L^p $ for $1 \leqq p \leqq \infty $, and that previous related results, which required the ordering cone to have a compact or weak-compact base, are not applicable in many normed vector lattices, including $C[a,b]$, $l^p $, and $L^p $ for $1 \leqq p \leqq \infty $.

Proper Efficient Points for Maximizations with Respect to Cones

Proper efficient points (Pareto maxima) are defined in tangent cone terms and are characterized by the existence of equivalent real-valued maximization problems.