Semistrictly quasiconvex mappings and non-convex vector optimization

This paper introduces a new class of non-convex vector functions strictly larger than that of P-quasiconvexity, with P ^m being the underlying ordering cone, called semistrictly ( ^m\ –int P)-quasiconvex functions. This notion allows us to unify various results on existence of weakly efficient (weakly Pareto) optima. By imposing a coercivity condition we establish also the compactness of the set of weakly Pareto solutions. In addition, we provide various characterizations for the non-emptiness, convexity and compactness of the solution set for a subclass of quasiconvex vector optimization problems on the real-line. Finally, it is also introduced the notion of explicit ( ^m\ –int P)-quasiconvexity (equivalently explicit (int P)-quasiconvexity) which plays the role of explicit quasiconvexity (quasiconvexity and semistrict quasiconvexity) of real-valued functions.Acknowldegements.The author wishes to thank both referees for their careful reading of the paper, their comments, remarks, helped to improve the presentation of some results. One of the referee provided the references [5, 6] and indirectly [20].     

Existence of positive solutions of nonlinear fractional differential equations

Existence of positive solutions for the nonlinear fractional differential equation D^su(x)=f(x,u(x)), 0<s<1, has been studied (S. Zhang, J. Math. Anal. Appl. 252 (2000) 804-812), where D^s denotes Riemann-Liouville fractional derivative. In the present work we study existence of positive solutions in case of the nonlinear fractional differential equation:

L(D)u=f(x,u),u(0)=0,0<x<1,     

Queueing Dynamics and Maximal Throughput Scheduling in Switched Processing Systems

We study a processing system comprised of parallel queues, whose individual service rates are specified by a global service mode (configuration). The issue is how to switch the system between various possible service modes, so as to maximize its throughput and maintain stability under the most workload-intensive input traffic traces (arrival processes). Stability preserves the job inflow–outflow balance at each queue on the traffic traces. Two key families of service policies are shown to maximize throughput, under the mild condition that traffic traces have long-term average workload rates. In the first family of cone policies, the service mode is chosen based on the system backlog state belonging to a corresponding cone. Two distinct policy classes of that nature are investigated, MaxProduct and FastEmpty. In the second family of batch policies (BatchAdapt), jobs are collectively scheduled over adaptively chosen horizons, according to an asymptotically optimal, robust schedule. The issues of nonpreemptive job processing and non-negligible switching times between service modes are addressed. The analysis is extended to cover feed-forward networks of such processing systems/nodes. The approach taken unifies and generalizes prior studies, by developing a general trace-based modeling framework (sample-path approach) for addressing the queueing stability problem. It treats the queueing structure as a deterministic dynamical system and analyzes directly its evolution trajectories. It does not require any probabilistic superstructure, which is typically used in previous approaches. Probability can be superposed later to address finer performance questions (e.g., delay). The throughput maximization problem is seen to be primarily of structural nature. The developed methodology appears to have broader applicability to other queueing systems.     

Additive stable solutions on perfect cones of cooperative games

Closed kernel systems of the coalition matrix turn out to correspond to cones of games on which the core correspondence is additive and on which the related barycentric solution is additive, stable and continuous. Different perfect cones corresponding to closed kernel systems are described. Received: December 2001/Revised: July 2002 RID="*" ID="*"  This note contains the new results, which were presented by the first author in an invited lecture at the XIV Italian Meeting on Game Theory and Applications in Ischia, July 2001. The lecture was dedicated to Irinel Dragan on the occasion of his seventieth birthday.     

Vector Equilibrium Problems Under Asymptotic Analysis

Given a closed convex set K in Rⁿ; a vector function F:K×K R^m; a closed convex (not necessarily pointed) cone P(x) in ^m with non-empty interior, PP(x) Ø, various existence results to the problemfind xK such that F(x,y)- int P(x) y K under P(x)-convexity/lower semicontinuity of F(x,) and pseudomonotonicity on F, are established. Moreover, under a stronger pseudomonotonicity assumption on F (which reduces to the previous one in case m=1), some characterizations of the non-emptiness of the solution set are given. Also, several alternative necessary and/or sufficient conditions for the solution set to be non-empty and compact are presented. However, the solution set fails to be convex in general. A sufficient condition to the solution set to be a singleton is also stated. The classical case P(x)=^m ₊ is specially discussed by assuming semi-strict quasiconvexity. The results are then applied to vector variational inequalities and minimization problems. Our approach is based upon the computing of certain cones containing particular recession directions of K and F.     

Normal cones of monomial primes

We explicitly calculate the normal cones of all monomial primes which define the curves of the form , where . All of these normal cones are reduced and Cohen-Macaulay, and their reduction numbers are independent of the reduction. These monomial primes are new examples of integrally closed ideals for which the product with the maximal homogeneous ideal is also integrally closed.

Substantial use was made of the computer algebra packages Maple and Macaulay2.

     

Continuation method for -sublinear mappings

Let be a real Banach space partially ordered by a closed convex cone with nonempty interior . We study the continuation method for the monotone operator which satisfies

for all , , where . Thompson's metric is among the main tools we are using.     

Some Geometry of the Cone of Nonnegative Definite Matrices and Weights of Associated X2 Distribution

Consider the test problem about matrix normal mean M with the null hypothesis M = O against the alternative that M is nonnegative definite. In our previous paper (Kuriki (1993, Ann. Statist., 21, 1379–1384)), the null distribution of the likelihood ratio statistic has been given in the form of a finite mixture of ² distributions referred to as X² distribution. In this paper, we investigate differential-geometric structure such as second fundamental form and volume element of the boundary of the cone formed by real nonnegative definite matrices, and give a geometric derivation of this null distribution by virtue of the general theory on the X² distribution for piecewise smooth convex cone alternatives developed by Takemura and Kuriki (1997, Ann. Statist., 25, 2368–2387).     

Asymptotic exponential cones of Metzler matrices and their use in the solution of an algebraic problem

The aim of this paper is that of investigating the asymptotic exponential cone of a single Metzler matrix, introduced in [23], and of defining and analysing the new concept of asymptotic exponential cone of a family of Metzler matrices (along a certain direction). These results will provide necessary and/or sufficient conditions for the solvability of an interesting algebraic problem that arises in the context of continuous-time positive switched systems and, specifically, in the investigation of the reachability property [21,22,25].     

SUPER EFFICIENCY AND ITS SCALARIZATION IN TOPOLOGICAL VECTOR SPACE

1. Introduction and PreliminariesRecently, Borwein and Zhuang[1,21 introduced the concept of super efficiency in normedlinear space. Super efficiency refines the notion of efficiency and other kinds of properefficiency; they provided concise scalar characterizations and duality results when the underlying decision problem is convex. They also established a Lagrange Multiplier Theoremfor super efficiency in convex settings and expressed super efficient points as saddle pointsof appropriate L…