On a domination property for vector maximization with respect to cones

We give a corrected version of Theorem 2.1 of Ref. 1.The author is indebted to D. T. Luc, Hungarian Academy of Sciences, Budapest, Hungary, for pointing out his error in the proof of the original version of Theorem 2.1 in Ref. 1.     

Efficiency and proper efficiency in vector maximization with respect to cones

Vector maximization problems arise when more than one objective function is to be maximized over a given feasibility region. The concepts of efficiency and proper efficiency have played a useful role in the analysis of such problems. Recently these concepts have been extended to vector maximization problems in which the underlying domination cone is a convex cone. In this paper, efficient and properly efficient solutions for the vector maximization problem in which the underlying domination cone is any nontrivial, closed convex cone are examined. Differences between properly and improperly efficient solutions are established. Characterizations of efficient and properly efficient solutions are presented, and conditions under which efficient solutions exist and fail to exist are derived.     

Recession cones and the domination property in vector optimization

This paper presents a study of recession cones of nonconvex sets in infinite dimensional spaces. The results are then applied to investigate efficiency conditions and the domination property in vector optimization.This paper was written when the author was at the University of Erlangen-Nürnberg under a grant of the Alexander von Humboldt-Stiftung.On leave from the Institute of Mathematics, Hanoi, Vietnam.     

On the domination property in vector optimization

An example is given to show the inadequacy of the result of Ref. 1 concerning the domination property of a convex vector maximization problem with respect to cones. A necessary and sufficient condition for the domination property to hold is supplied.     

On optimality conditions for maximizations with respect to cones

For a Pareto maximization problem defined in infinite dimensions in terms of cones, relationships among several types of maximal elements are noted and optimality conditions are developed in terms of tangent cones.     

On multi-transitivity with respect to a vector

A topological dynamical system(X,f)is said to be multi-transitive if for every n∈N the system(Xn,f×f2××fn)is transitive.We introduce the concept of multi-transitivity with respect to a vector and show that multi-transitivity can be characterized by the hitting time sets of open sets,answering a question proposed by Kwietniak and Oprocha(2012).We also show that multi-transitive systems are Li-Yorke chaotic.     

Proper efficiency with respect to cones

Strict separation by a cone is used here to redefine proper efficiency. Two versions of the properness, which unify and generalize known definitions, are presented. Necessary and sufficient conditions for the existence of the set of properly efficient decisions and characterization of this set in terms of the supports of the decision set are given.The research was done while the author was a visiting professor at the University of British Columbia, Vancouver, British Columbia, Canada.     

On integration of vector functions with respect to vector measures

We study integration of Banach space-valued functions with respect to Banach space-valued measures. We focus our attention on natural extensions to this setting of the Birkhoff and McShane integrals. The corresponding generalization of the Birkhoff integral was first considered by Dobrakov under the name S*-integral. Our main result states that S*-integrability implies McShane integrability in contexts in which the later notion is definable. We also show that a function is measurable and McShane integrable if and only if it is Dobrakov integrable (i.e. Bartle *-integrable).     

An existence result for maximizations with respect to cones

A sufficient condition is given for the existence of a solution to a generalized Pareto maximization problem in which maximization is defined in terms of cones. This result generalizes the fact that an upper semicontinuous real-valued function achieves its maximum on a compact set.     

Connected domination critical graphs with respect to relative complements

A dominating set in a graph G is a connected dominating set of G if it induces a connected subgraph of G. The minimum number of vertices in a connected dominating set of G is called the connected domination number of G, and is denoted by γ _c (G). Let G be a spanning subgraph of K _s,s and let H be the complement of G relative to K _s,s ; that is, K _s,s = G ⊕ H is a factorization of K _s,s . The graph G is k-γ _c -critical relative to K _s,s if γ _c (G) = k and γ _c (G + e) < k for each edge e ∈ E(H). First, we discuss some classes of graphs whether they are γ _c -critical relative to K _s,s . Then we study k-γ _c -critical graphs relative to K _s,s for small values of k. In particular, we characterize the 3-γ _c -critical and 4-γ _c -critical graphs.     

On well-posedness for parametric vector quasiequilibrium problems with moving cones

In this paper we consider weak and strong quasiequilibrium problems with moving cones in Hausdorff topological vector spaces. Sufficient conditions for well-posedness of these problems are established under relaxed continuity assumptions. All kinds of wellposedness are studied: (generalized) Hadamard well-posedness, (unique) well-posedness under perturbations. Many examples are provided to illustrate the essentialness of the imposed assumptions. As applications of the main results, sufficient conditions for lower and upper bounded equilibrium problems and elastic traffic network problems to be well-posed are derived.     

Stability of exponential utility maximization with respect to market perturbations

We investigate the continuity of expected exponential utility maximization with respect to perturbation of the Sharpe ratio of markets. By focusing only on continuity, we impose weaker regularity conditions than those found in the literature. Specifically, we require, in addition to the V$V$-compactness hypothesis of Larsen and ?itkovi? (2007) [13], a local bmo$bmo$ hypothesis, a condition which is essentially implicit in the setting of [13]. For markets of the form S=M+∫λd〈M〉$S=M+\int \lambda d〈M〉$, these conditions are simultaneously implied by the existence of a uniform bound on the norm of λ⋅M$\lambda \cdot M$ in a suitable bmo$bmo$ space.     

Existence and characterization of efficient decisions with respect to cones

Existence and characterization theorems for the efficient (nondominated) set of decisions inR ⁿ are presented. The existence is proved when the set of decisions satisfies some compactness conditions. The efficient set is characterized in terms of the exposed efficient decisions when certain convexity and compactness conditions are satisfied.     

The limit case of a domination property

The domination number γ(G) of a connected graph G of order n is bounded below by(n+2-e(G))/ 3 , where (G) denotes the maximum number of leaves in any spanning tree of G. We show that (n+2-e(G))/ 3 = γ(G) if and only if there exists a tree T ∈ T ( G) ∩ R such that n1(T ) = e(G), where n1(T ) denotes the number of leaves of T1, R denotes the family of all trees in which the distance between any two distinct leaves is congruent to 2 modulo 3, and T (G) denotes the set composed by the spanning trees of G. As a consequence of the study, we show that if (n+2-e(G))/ 3 = γ(G), then there exists a minimum dominating set in G whose induced subgraph is an independent set. Finally, we characterize all unicyclic graphs G for which equality (n+2-e(G))/ 3= γ(G) holds and we show that the length of the unique cycle of any unicyclic graph G with (n+2-e(G))/ 3= γ(G) belongs to {4} ∪ {3 , 6, 9, . . . }.     

On integration with respect to a DT-measure

DT-measures generalize the topological measures of Aarnes. We continue to study an integral of a real-valued continuous function on compact Hausdorff space with respect to a DT-measure and investigate the following: multiplicativity of this integral, \(w*\)-topology on the family of DT-measures and density of various subfamilies of this space, and also integral as a set function.