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.     

Tangent cones and Dini derivatives

We prove a property of the Bouligand tangent cone to the epigraph (or to the graph) of a locally Lipschitz function. It is also shown how this result can be used in determining Dini sequences. Finally, some relationships between such a cone and Dini derivatives are provided.     

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.The research was conducted while the author was working on his PhD Degree under the supervision of Professor J. M. Borwein, whose guidance and valuable suggestions are gratefully appreciated. The author would like to thank two anonymous referees for their constructive comments and suggestions. This research was supported by an NSERC grant and a Mount Saint Vincent University Research Grant.     

Locally convex quotient lattice cones

Walter Roth has investigated certain equivalence relations on locally convex cones in [W. Roth, Locally convex quotient cones, J. Convex Anal. 18, No. 4, 903–913 (2011)] which give rise to the definition of a locally convex quotient cone. In this paper, we investigate some special equivalence relations on a locally convex lattice cone by which the locally convex quotient cone becomes a lattice. In the case of a locally convex solid Riesz space, this reduces to the known concept of locally convex solid quotient Riesz space. We prove that the strict inductive limit of locally convex lattice cones is a locally convex lattice cone. We also study the concept of locally convex complete quotient lattice cones.     

A uniform boundedness theorem for locally convex cones

We prove a uniform boundedness theorem for families of linear operators on ordered cones. Using the concept of locally convex cones we introduce the notions of barreled cones and of weak cone-completeness. Our main result, though no straightforward generalization of the classical case, implies the Uniform Boundedness Theorem for Fréchet spaces.

     

Remarks on convex cones

We point out in this note that the class of cones in a locally convex topological vector space satisfying property () or piecewise relatively weakly compact cones is exactly the class of cones admitting weakly compact bases or the class of cones whose closures admit weakly compact bases.This work was supported by a Monash University Postdoctoral Fellowship.     

A note on convex cones and constraint qualifications in infinite-dimensional vector spaces

In connection with mathematical programming in infinite-dimensional vector spaces, Zowe has studied the relationship between the Slater constraint qualification and a formally weaker qualification used by Kurcyusz. The attractive feature of the latter is that it involves only active constraints. Zowe has proved that, in barreled spaces, the two qualifications are equivalent and has asked whether the assumption of barreledness is superfluous. By studying cores and interiors of convex cones, we show that the two constraint qualifications are equivalent in a given topological vector spaceE iff every barrel inE is a neighborhood of the origin. Thus, whenE is locally convex, the two constraint qualifications are equivalent iffE is barreled. Other questions of Zowe are also answered.This research was supported in part by the Office of Naval Research, and in part by the Sonderforschungsbereich 21, Institut für Operations Research, Bonn, Federal Republic of Germany. The author is indebted to Professor J. Zowe for some helpful comments.     

Theory of convex cones in multicriteria decision making

We develop the theory of convex polyhedral cones in the objective-function space of a multicriteria decision problem. The convex cones are obtained from the decision-maker's pairwise judgments of decision alternatives and are applicable to any quasiconcave utility function. Therefore, the cones can be used in any progressively articulated solution procedure that employs pairwise comparisons. The cones represent convex sets of solutions that are inferior to known solutions to a multicriteria problem. Therefore, these convex sets can be eliminated from consideration while solving the problem. We develop the underlying theory and a framework for representing knowledge about the decision-maker's preference structure using convex cones. This framework can be adopted in the interactive solution of any multicriteria problem after taking into account the characteristics of the problem and the solution procedure. Our computational experience with different multicriteria problems shows that this approach is both viable and efficient in solving practical problems of moderate size.     

Philon's line generalized: An optimization problem from geometry

Consider inn-dimensional Euclidean space the intersection of a convex cone and a hyperplane through a given point. The problem is to minimize the (n-1)-volume of this intersection. A geometric interpretation of the first-order optimality condition is given. The special casen=2 is known as a characteristic property of Philon's line.Editorial Note. Because of Professor Wetterling's death on January 21, 1994, this paper was handled by Dr. F. Twilt, Department of Applied Mathematics University of Twente, Enschede, Netherlands.     

Local minima of quadratic forms on convex cones

We study the local minima and the critical values of a quadratic form on the trace of a convex cone. This variational problem leads to the development of a spectral theory that combines matrix algebra and facial analysis of convex cones.      

Variable preference modeling with ideal-symmetric convex cones

Based on the concept of general domination structures, this paper presents an approach to model variable preferences for multicriteria optimization and decision making problems. The preference assumptions for using a constant convex cone are given, and, in remedy of some immanent model limitations, a new set of assumptions is presented. The underlying preference model is derived as a variable domination structure that is defined by a collection of ideal-symmetric convex cones. Necessary and sufficient conditions for nondominance are established, and the problem of finding corresponding nondominated solutions is addressed and solved on examples.     

The single-server queue with uniformly limited actual waiting times

This paper studies a single-server queueing system in which no customer has to wait for a duration longer than a constant K. If the waiting time is longer than K, then the service time of the previous customer will have to be cut short. Using analytical method together with the property that the queueing process ‘starts anew’ probabilistically whenever an arriving customer initiates a busy period, we obtain various transient and stationary solutions for the system.     

Implicit complementarity problems on isotone projection cones

Isac and Németh [G. Isac and A. B. Németh, Projection methods, isotone projection cones and the complementarity problem, J. Math. Anal. Appl. 153 (1990), pp. 258–275] proved that solving a coincidence point equation (fixed point problem) in turn solves the corresponding implicit complementarity problem (nonlinear complementarity problem) and they exploited the isotonicity of the metric projection onto isotone projection cones to solve implicit complementarity problems (nonlinear complementarity problems) defined by these cones. In this article an iterative algorithm is studied in connection with an implicit complementarity problem. It is proved that if the sequence generated through the defined algorithm is convergent, then its limit is a solution of the coincidence point equation and thus solves the implicit complementarity problem. Sufficient conditions are given for this sequence to be convergent for implicit complementarity problems defined by isotone projection cones, extending the results of Németh [S.Z. Németh, Iterative methods for nonlinear complementarity problems on isotone projection cones, J. Math. Anal. Appl. 350 (2009), pp. 340–370]. Some existing concepts from the latter paper are extended to solve the problem of finding nonzero solutions of the implicit complementarity problem.     

Generalized tangent cone and an optimization problem in a normed space

Most abstract multiplier rules in the literature are based on the tangential approximation at a point to some set in a Banach space. The present paper is concerned with the study of a generalized tangent cone, which is a tangential approximation to that set at a common point of two sets. The new notion of tangent cone generalizes previous concepts of tangent cones. This generalized tangent cone is used to characterize the optimality conditions for a simultaneous maximization and minimization problem. The paper is of theoretical character; practical applications are not found so far.     

一类非光滑多目标规划问题的最优性条件

研究了一类非光滑多目标规划问题.这类多目标规划问题的目标函数为锥凸函数与可微函数之和,其约束条件是Euclidean空间中的锥约束.在满足广义Abadie约束规格下,利用广义Farkas引理和多目标函数标量化,给出了这一类多目标规划问题的锥弱有效解最优性必要条件.     

一类多目标广义分式规划问题的最优性条件和对偶

研究了一类不可微多目标广义分式规划问题．首先,在广义Abadie约束品性条件下,给出了其真有效解的Kuhn—Tucker型必要条件．随后,在（C,a,P,d）一凸性假设下给出其真有效解的充分条件．最后,在此基础上建立了一种对偶模型,证明了对偶定理．得到的结果改进了相关文献中的相应结论．     

Characterization of the barrier parameter of homogeneous convex cones

We characterize the smallest (best) barrier parameter of self-concordant barriers for homogeneous convex cones. In particular, we prove that this parameter is the same as the rank of the cone which is the number of steps in a recursive construction of the cone (Siegel domain construction). We also provide lower bounds on the barrier parameter in terms of the Carathéodory number of the cone. The bounds are tight for homogeneous self-dual cones. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.Research supported in part by an operating grant from NSERC of Canada.Research supported in part by the National Science Foundation under grant DMS-9306318.