Monotonicity and best approximation in Orlicz-Sobolev spaces with the Luxemburg norm

Criteria for strict monotonicity, upper (lower) locally uniform monotonicity and uniform monotonicity of Orlicz-Sobolev spaces with the Luxemburg norm are given. Some applications to best approximation are presented.     

Dual characterization of properties of risk measures on Orlicz hearts

We extend earlier representation results for monetary risk measures on Orlicz hearts. Then we give general conditions for such risk measures to be Gateaux-differentiable, strictly monotone with respect to almost sure inequality, strictly convex modulo translation, strictly convex modulo comonotonicity, or monotone with respect to different stochastic orders. The theoretical results are used to analyze various specific examples of risk measures. We thank Andreas Hamel and Michael Kupper for fruitful discussions and helpful comments. P. Cheridito has been supported by NSF Grant DMS-0642361, a Rheinstein Award and a Peek Fellowship. T. Li has been supported by a Marshall Scholarship and a Merage Fellowship.     

Perturbed kernel approximation on homogeneous manifolds

Current methods for interpolation and approximation within a native space rely heavily on the strict positive-definiteness of the underlying kernels. If the domains of approximation are the unit spheres in euclidean spaces, then zonal kernels (kernels that are invariant under the orthogonal group action) are strongly favored. In the implementation of these methods to handle real world problems, however, some or all of the symmetries and positive-definiteness may be lost in digitalization due to small random errors that occur unpredictably during various stages of the execution. Perturbation analysis is therefore needed to address the stability problem encountered. In this paper we study two kinds of perturbations of positive-definite kernels: small random perturbations and perturbations by Dunkl's intertwining operators [C. Dunkl, Y. Xu, Orthogonal polynomials of several variables, Encyclopedia of Mathematics and Its Applications, vol. 81, Cambridge University Press, Cambridge, 2001]. We show that with some reasonable assumptions, a small random perturbation of a strictly positive-definite kernel can still provide vehicles for interpolation and enjoy the same error estimates. We examine the actions of the Dunkl intertwining operators on zonal (strictly) positive-definite kernels on spheres. We show that the resulted kernels are (strictly) positive-definite on spheres of lower dimensions.     

Portfolio selection under strict uncertainty: A multi-criteria methodology and its application to the Frankfurt and Vienna Stock Exchanges

In modern portfolio theory, it is common practice to first compute the risk-reward efficient frontier and then to support an individual investor in selecting a portfolio that meets his/her preferences for profitability and risk. Potential flaws include (a) the assumption that past data provide sufficient evidence for predicting the future performances of the securities under consideration and (b) the necessity to mathematically determine or approximate the investor’s utility function. In this paper, we propose a methodology whose initial phase filters portfolios that are inefficient from a historical perspective. While this is consistent with traditional approaches, the second phase differs from the standard approach as it uses a decision table constructed by considering multiple scenarios assuming strict uncertainty. The table cells measure consequences by a multi-criteria linear performance index of simulated future returns, which avoids difficulties with performance ratios. The real world applicability is illustrated through two studies based on data from the stock exchanges in Frankfurt and Vienna.     

Equivalence of Two Nondegeneracy Conditions for Semidefinite Programs

Nondegeneracy assumptions are often needed in order to prove the local fast convergence of suitable algorithms as well as in the sensitivity analysis for semidefinite programs. One of the more standard nondegeneracy conditions is the geometric condition used by Alizadeh et al. (Math. Program. 77:111–128, 1997). On the other hand, Kanzow and Nagel (SIAM J. Optim. 15:654–672, 2005) recently introduced an algebraic condition that was used in order to prove, for the first time, the local quadratic convergence of a suitable algorithm for the solution of semidefinite programs without using the strict complementarity assumption. The aim of this paper is to show that these two nondegeneracy conditions are equivalent.     

A General Duality Scheme for Nonconvex Minimization Problems with a Strict Inequality Constraint

We establish a duality formula for the problem Minimize f(x)+g(x) for h(x)+k(x)<0 where g, k are extended-real-valued convex functions and f, h belong to the class of functions that can be written as the lower envelope of an arbitrary family of convex functions. Applications in d.c. and Lipschitzian optimization are given.     

3一致 C-超图的最小边数

混合超图是含有两类超边的超图,一类称为C-超边,一类称为D-超边,它们的区别主要体现在染色要求上．混合超图的染色,要求每一C-超边至少有两个点染相同的颜色,而每一D-超边至少有两个点染不同的颜色．所用的最大颜色数称为对应混合超图的上色数,所用的最小颜色数称为对应混合超图的下色数．上、下色数与边数有密切关系．作者在文献[2]中证明了具有最小上色数的3一致C-超图边数的一个下界为‘n(n-2)/3’,其中n为对应混合超图的顶点数．该文证明当n=2k 1时,该下界是可以达到的．     

基于模糊蕴涵的分配性方程

研究函数方程组I(x,T(y,z))=T(I(x,y),I(x,z)),I(x,y)=I(N(y),N(x))的解,其中T:[0,1]2→[0,1]是一个严格三角模,I:[0,1]2→[0,1]是一个模糊蕴涵算子和N:[0,1]→[0,1]是一个强否定.在I除了在点(0,0),(1,1)不连续的假设下,获得了满足这个函数方程组解的完全刻画.