Existence of maximal points with respect to ordered bipreference relations

The maximality of a point with respect to an ordered pair of arbitrary preference relations is introduced. Necessary and sufficient conditions for the existence of maximal points are given.The authors are grateful for the referee's comments which led to an improved presentation of the paper.     

Conjugate Intervals for the Linear Fixed-Endpoint Control Problem

For certain optimal control problems with piecewise continuous controls, recently Loewen and Zheng (Ref. 1) and Zeidan (Ref. 2) defined two sets of generalized conjugate points for which, under normality assumptions, the second-order conditions in terms of the accessory problem imply their emptiness. However, simple examples show that checking the existence of such points may be more difficult than directly finding variations that make the second variation negative. In this paper, for the linear fixed-endpoint control problem, we introduce a new set whose emptiness is equivalent to the nonnegativity of the second variation along admissible variations. Moreover, we achieve by means of this set the main objective of introducing a characterization of this condition, namely, to obtain a simpler way of verifying it.     

Sharply transitive sets in quasigroup actions

This paper forms part of the general development of the theory of quasigroup permutation representations. Here, the concept of sharp transitivity is extended from group actions to quasigroup actions. Examples of nontrivial sharply transitive sets of quasigroup actions are constructed. A general theorem shows that uniformity of the action is necessary for the existence of a sharply transitive set. The concept of sharp transitivity is related to two pairwise compatibility relations and to maximal cliques within the corresponding compatibility graphs.     

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.     

Stability of solutions for variational relation problems with applications

In this paper, we prove that most of problems in variational relations (in the sense of Baire category) are essential and that, for any problem in variational relations, there exists at least one essential component of its solution set. As applications, we deduce the existence of essential components of the set of Ky Fan’s points based on Ky Fan’s minimax inequality theorem, the existence of essential components of the set of Nash equilibrium points for general n-person non-cooperative games, the existence of essential component of the set of solutions for vector Ky Fan’s minimax inequality, the existence of essential components of the set of KKM points and the existence of essential components of the set of solutions for Ky Fan’s section theorem.     

Minimax Results and Finite-Dimensional Separation

In this paper, we review and unify some classes of generalized convex functions introduced by different authors to prove minimax results in infinite-dimensional spaces and show the relations between these classes. We list also for the most general class already introduced by Jeyakumar (Ref. 1) an elementary proof of a minimax result. The proof of this result uses only a finite-dimensional separa- tion theorem; although this minimax result was already presented by Neumann (Ref. 2) and independently by Jeyakumar (Ref. 1), we believe that the present proof is shorter and more transparent.     

Generalization of an Existence Theorem for Variational Inequalities

By using the concept of exceptional family of elements, Zhao proposed a new existence theorem for variational inequalities over a general nonempty closed convex set (Ref. 1, Theorem 2.3), which is a generalization of the well-known Moré's existence theorem for nonlinear complementarity problems. The proof of Theorem 2.3 in Ref. 1 depends strongly on the condition 0∈K. Since this condition is rather strict for a general variational inequality, Zhao proposed an open question at the end of Ref. 1: Can the condition 0∈K in Theorem 2.3 be removed? In this paper, we answer this open question. Furthermore, we present the new notion of exceptional family of elements and establish a theorem of the alternative, by which we develop two new existence theorems for variational inequalities. Our results generalize the Zhao existence result.     

拟弱几乎周期点的等价定义与系统的混沌性

1992年, 周作领引进了弱几乎周期点这一概念. 1995年, 周和何伟弘又引进了拟弱几乎周期点这个概念, 并利用它们深刻地刻画了一个动力系统的本质所在. 为了更好地看出这两者的区别,首先从回复时间集的角度给出拟弱几乎周期点的等价定义,然后研究了一个存在真的拟弱几乎周期点的系统的混沌情况,得到了这样的系统是Takens-Ruelle混沌的.     

Collective fixed points and maximal elements with applications to abstract economies

In this paper, we first establish collective fixed points theorems for a family of multivalued maps with or without assuming that the product of these multivalued maps is Φ-condensing. As an application of our collective fixed points theorem, we derive the coincidence theorem for two families of multivalued maps defined on product spaces. Then we give some existence results for maximal elements for a family of L_S-majorized multivalued maps whose product is Φ-condensing. We also prove some existence results for maximal elements for a family of multivalued maps which are not L_S-majorized but their product is Φ-condensing. As applications of our results, some existence results for equilibria of abstract economies are also derived. The results of this paper are more general than those given in the literature.     

Vector Variational Inequalities Involving Vector Maximal Points

This paper considers the existence of solutions and the equivalence of four kinds of vector variational inequalities (VVI). More precisely, a sufficient condition is provided under which the solution sets of these VVIs are nonempty and equal. An example is given, showing that such a sufficient condition is essential to ensure the results. Actually, the main theorems in this paper can be regarded as a suitable correction and a refinement of recent results due to Chang et al. (Ref. 1).     

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.     

On topological existence theorems and applications to optimization-related problems

In this paper, we establish a continuous selection theorem and use it to derive five equivalent results on the existence of fixed points, sectional points, maximal elements, intersection points and solutions of variational relations, all in topological settings without linear structures. Then, we study the solution existence of a number of optimization-related problems as examples of applications of these results: quasivariational inclusions, Stampacchia-type vector equilibrium problems, Nash equilibria, traffic networks, saddle points, constrained minimization, and abstract economies.     

Characterizing efficiency without linear structure: a unified approach

In this paper, we study a general optimization problem without linear structure under a reflexive and transitive relation on a nonempty set E, and characterize the existence of efficient points and the domination property for a subset of E through a generalization of the order-completeness condition introduced earlier. Afterwards, we study the abstract optimization problem by using generalized continuity concepts and establish various existence results. As an application, we extend and improve several existence results given in the literature for an optimization problem involving set-valued maps under vector and set criteria. The work of the first author was partially supported by CONICYT-Chile through FONDECYT 107-0689 and FONDAP-Matemáticas Aplicadas II; whereas that of the second and third author were supported in part by Ministerio de Educación y Ciencia (Spain), project MTM2006-02629 and by Junta de Castilla y León (Spain) Project VA027B06.     

NP-hardness results for the aggregation of linear orders into median orders

Given a collection Π of individual preferences defined on a same finite set of candidates, we consider the problem of aggregating them into a collective preference minimizing the number of disagreements with respect to Π and verifying some structural properties. We study the complexity of this problem when the individual preferences belong to any set containing linear orders and when the collective preference must verify different properties, for instance transitivity. We show that the considered aggregation problems are NP-hard for different types of collective preferences (including linear orders, acyclic relations, complete preorders, interval orders, semiorders, quasi-orders or weak orders), if the number of individual preferences is sufficiently large.     

Stochastic equations on compact groups in discrete negative time

In this paper a stochastic equation on compact groups in discrete negative time is studied. The diagonal group action on the extreme points of solutions is proved to be transitive by means of the coupling method. This result is applied to generalize Yor’s work which is closely related to Tsirelson’s stochastic differential equation and to give criteria for existence of a strong solution and for uniqueness in law. This research was supported by Open Research Center Project for Private Universities: matching fund subsidy from MEXT, 2004–2008.     

NoteA 16-vertex tournament for which Banks set and Slater set are disjoint

Given a tournament T, a Banks winner of T is the first vertex of any maximal (with respect to inclusion) transitive subtournament of T; a Slater winner of T is the first vertex of any transitive tournament at minimum distance of T (the distance being the number of arcs to reverse in T to make T transitive). In this note, we show that there exists a tournament with 16 vertices for which no Slater winner is a Banks winner. This counterexample improves the previous one, due to G. Laffond and J.-F. Laslier, which has 75 vertices.     

Maximal points with respect to cone dominance in Banach spaces and their existence

Given a convex cone in a Banach spaceV, an examination of the cone maximal points of a setX inV (with respect to the cone dominance induced by ) with respect to their characterization and existence is undertaken. The totality of cone maximal points ofX is called the conical frontier ofX. Comparisons of the conical frontiers of related sets and corresponding to related cones are made. By relaxing the compactness requirements of the underlying setX and by assuming some cone-related weaker forms of compactness, existence theorems for cone maximal points are developed. These theorems are believed to be generalizations of the existing results in one way or another.Maximizing points onX of certain linear functionals in the dual cone * of provide natural examples of cone maximal points. Properties characterizing a maximizing point of a linear functional in *, including the generalized version of Geoffrion's characterization of proper efficiency, are compiled and proved to be valid characterizations. Functionals in * with special properties are studied. Existence theorems are also obtained for the maximizing points of these functionals.The author is indebted to Professor James V. Whittaker for helpful discussions and comments and to Professors P. S. Bullen, C. W. Clark, B. N. Moyls, and F. Y. M. Wan for their encouragement and support.     

Banach空间中高阶常微分方程初边值问题的可解性

利用基于度理论的不动点定理,该文给出了Ｂａｎａｃｈ空间中ｎ阶常微分方程的初、边值问题有解的某些充分条件,同时证明了一类最大最小解的存在性．推广了现有文献中的某些结果．     

Essential Stability of Solutions for Maximal Element Theorem with Applications

In this paper, we prove that most of problems in maximal element theorem (in the sense of Baire category) are essential and that, for any problem in maximal element theorem, there exists at least one essential component of its solution set. As applications, we deduce the existence of essential components of the set of Ky Fan’s points based on Ky Fan Minimax Inequality, the existence of essential components of the set of Nash equilibrium points for general n-person non-cooperative games and the existence of essential components of the set of solutions of vector Ky Fan Minimax Inequality.     

An exponential penalty method for nondifferentiable minimax problems with general constraints

A well-known approach to constrained minimization is via a sequence of unconstrained optimization computations applied to a penalty function. This paper shows how it is possible to generalize Murphy's penalty method for differentiable problems of mathematical programming (Ref. 1) to solve nondifferentiable problems of finding saddle points with constraints. As in mathematical programming, it is shown that the method has the advantages of both Fiacco and McCormick exterior and interior penalty methods (Ref. 2). Under mild assumptions, the method has the desirable property that all trial solutions become feasible after a finite number of iterations. The rate of convergence is also presented. It should be noted that the results presented here have been obtained without making any use of differentiability assumptions.