基于同单调理论的IBNR准备金估计的随机界

在对IBNR准备金估计的随机模型存在的不足进行分析的基础上,对采用二阶段广义线性模型得到的各单元IBNR准备金估计和的分布问题进行讨论.在考虑折现因子的基础上,采用同单调理论,得到各单元准备金估计的和在凸序意义下的随机界,并通过一个实例对所述方法进行验证.     

凸多乘积问题的完全多项式时间近似算法

针对凸多乘积问题,提出一种求其全局最优解的近似算法.首先,通过引入参量获得一个等价问题,然后估计问题中每一乘积项的上下界,进而借助网格结点,获得一些凸规划问题,通过求解这些凸规划问题获得原问题的近似最优解.最后,给出了该算法的收敛性证明和计算复杂性分析.     

伪凸集值映射的误差界

考虑了伪凸集值映射的误差界.证明了对于伪凸集值映射,局部误差界成立意味着整体误差界成立.通过相依导数,给出了伪凸集值映射存在误差界的一些等价叙述.     

关于高斯低通滤波器的超分辨分析

对于经过高斯低通滤波的信号,通过求解一类凸优化模型稳定地恢复该信号的高频信息.当信号满足一定的分离条件时,给出了误差估计的界,从理论上证明了求解凸优化方法的稳定性.理论的证明依赖于压缩感知中的对偶理论.一个显著的差异在于高斯低通滤波器并不满足压缩感知中对于测量矩阵的要求,例如相关性,约束等距性质等.     

调和函数类的卷积的凸半径估计

利用调和函数的偏差性质、系数估计等方法对调和映照类的卷积的凸半径进行了深入研究,得到了一系列精确的结论.此外,通过选取不同的参数值得到凸半径与参数之间的关系.     

非线性自回归序列L1估计的渐近分布

由于时间序列数据中经常出现的厚尾特征使得通常的估计方法不再具有渐近的正态分布,在误差项二阶矩有限的条件下考虑了非线性自回归序列的L_1估计.采用局部线性近似的方法得到了具有凸样本路径的随机过程,在此基础上利用凸样本路径随机过程弱收敛的性质证明了非线性自回归序列L_1估计的渐近正态性及无偏性.     

Banach空间中的广义Dunkl-Williams常数及其性质

本文给出了广义Dunkl-Williams常数与一些著名几何常数例如凸系数、光滑系数、James常数之间的关系,从而得到一些蕴含不动点性质的充分条件,另外通过广义Dunkl-Williams常数的上下界的估计给出了Banach空间一致非方的刻画.     

一类相对于A的螺形映照的偏差上界估计

偏差估计一直是多复变函数研究的热点之一.近几年,双全纯凸(准凸)映照的偏差估计已经被研究人员估计出来,但螺形映照及其子类的偏差估计结果还为数不多.针对这一点,通过使用相对于A的螺形映照的定义及定义中已知的不等式,我们得到了一类相对于A的螺形映照的偏差上界估计.     

严格双对角占优矩阵行列式的上下界估计

严格双对角占优矩阵的行列式计算是数值代数中的热点问题. 本文首先将严格双对角占优矩阵右乘一个正对角矩阵, 使其化为严格对角占优矩阵, 其次对严格对角占优矩阵行列式的上下界进行估计, 从而得到严格双对角占优矩阵行列式的上下界估计. 最后通过数值算例表明所得估计是有效的.     

随机删失数据下核密度估计的Berry-Esseen界

本文在随机删失数据下研究了概率密度函数的核估计,获得了此核估计的一个Berry－Esseen界.     

A new variable reduction technique for convex integer quadratic programs

Convex integer quadratic programming involves minimization of a convex quadratic objective function with affine constraints and is a well-known NP-hard problem with a wide range of applications. We proposed a new variable reduction technique for convex integer quadratic programs (IQP). Based on the optimal values to the continuous relaxation of IQP and a feasible solution to IQP, the proposed technique can be applied to fix some decision variables of an IQP simultaneously at zero without sacrificing optimality. Using this technique, computational effort needed to solve IQP can be greatly reduced. Since a general convex bounded IQP (BIQP) can be transformed to a convex IQP, the proposed technique is also applicable for the convex BIQP. We report a computational study to demonstrate the efficacy of the proposed technique in solving quadratic knapsack problems.     

Discussions on non-probabilistic convex modelling for uncertain problems

Non-probabilistic convex model utilizes a convex set to quantify the uncertainty domain of uncertain-but-bounded parameters, which is very effective for structural uncertainty analysis with limited or poor-quality experimental data. To overcome the complexity and diversity of the formulations of current convex models, in this paper, a unified framework for construction of the non-probabilistic convex models is proposed. By introducing the correlation analysis technique, the mathematical expression of a convex model can be conveniently formulated once the correlation matrix of the uncertain parameters is created. More importantly, from the theoretic analysis level, an evaluation criterion for convex modelling methods is proposed, which can be regarded as a test standard for validity verification of subsequent newly proposed convex modelling methods. And from the practical application level, two model assessment indexes are proposed, by which the adaptabilities of different convex models to a specific uncertain problem with given experimental samples can be estimated. Four numerical examples are investigated to demonstrate the effectiveness of the present study.     

Discretize-then-relax approach for convex/concave relaxations of the solutions of parametric ODEs

This paper presents a discretize-then-relax methodology to compute convex/concave bounds for the solutions of a wide class of parametric nonlinear ODEs. The procedure builds upon interval methods for ODEs and uses the McCormick relaxation technique to propagate convex/concave bounds. At each integration step, a two-phase procedure is applied: a priori convex/concave bounds that are valid over the entire step are calculated in the first phase; then, pointwise-in-time convex/concave bounds at the end of the step are obtained in the second phase. An approach that refines the interval state bounds by considering subgradients and affine relaxations at a number of reference parameter values is also presented. The discretize-then-relax method is implemented in an object-oriented manner and is demonstrated using several numerical examples.     

Calibration and convex programming: two approaches to one problem

Calibration is a common technique of data processing in survey sampling. Although convex programming would be an obvious tool for this purpose, usually other methods are used in the practice of statistical institutes. A comparison of those methods and convex programming is reported on in this paper.     

Reduction of quasidifferentials and minimal representations

Some criterias for the non-minimality of pairs of compact convex sets of a real locally convex topological vector space are proved, based on a reduction technique via cutting planes and excision of compact convex subsets. Following an example of J. Grzybowski, we construct a class of equivalent minimal pairs of compact convex sets which are not connected by translations.Corresponding author.     

Anti-optimization technique – a generalization of interval analysis for nonprobabilistic treatment of uncertainty

Anti-optimization technique, on the one hand, represents an alternative and complement to traditional probabilistic methods, and on the other hand, it is a generalization of the mathematical theory of interval analysis. In this study, in terms of interval analysis or interval mathematics, the arithmetic operations and the partial order relation of anti-optimization technique can be defined, and the convex model variables and the convex model extension function of convex models can also be introduced. The comparison of the Lagrange multiplier method with the convex model extension method for evaluating the region of static displacements of structures with uncertain-but-bounded parameters shows that the width of the upper and lower bounds on the static displacement yielded by the Lagrange multiplier method of convex models is tighter than those produced by the convex model extension.     

A Polyhedral Approach for Nonconvex Quadratic Programming Problems with Box Constraints

We apply a linearization technique for nonconvex quadratic problems with box constraints. We show that cutting plane algorithms can be designed to solve the equivalent problems which minimize a linear function over a convex region. We propose several classes of valid inequalities of the convex region which are closely related to the Boolean quadric polytope. We also describe heuristic procedures for generating cutting planes. Results of preliminary computational experiments show that our inequalities generate a polytope which is a fairly tight approximation of the convex region.     

一类线性乘积规划问题的分支定界缩减方法

提出了求解一类线性乘积规划问题的分支定界缩减方法, 并证明了算法的收敛性.在这个方法中, 利用两个变量乘积的凸包络技术, 给出了目标函数与约束函数中乘积的下界, 由此确定原问题的一个松弛凸规划, 从而找到原问题全局最优值的下界和可行解. 为了加快所提算法的收敛速度, 使用了超矩形的缩减策略. 数值结果表明所提出的算法是可行的.     

Convex extensions and envelopes of lower semi-continuous functions

 We define a convex extension of a lower semi-continuous function to be a convex function that is identical to the given function over a pre-specified subset of its domain. Convex extensions are not necessarily constructible or unique. We identify conditions under which a convex extension can be constructed. When multiple convex extensions exist, we characterize the tightest convex extension in a well-defined sense. Using the notion of a generating set, we establish conditions under which the tightest convex extension is the convex envelope. Then, we employ convex extensions to develop a constructive technique for deriving convex envelopes of nonlinear functions. Finally, using the theory of convex extensions we characterize the precise gaps exhibited by various underestimators of $x/y$ over a rectangle and prove that the extensions theory provides convex relaxations that are much tighter than the relaxation provided by the classical outer-linearization of bilinear terms. Received: December 2000 / Accepted: May 2002 Published online: September 5, 2002 RID="*" ID="*" The research was funded in part by a Computational Science and Engineering Fellowship to M.T., and NSF CAREER award (DMI 95-02722) and NSF/Lucent Technologies Industrial Ecology Fellowship (NSF award BES 98-73586) to N.V.S. Key words. convex hulls and envelopes – multilinear functions – disjunctive programming – global optimization     

On the mean number of normals through a point in the interior of a convex body

Recently, Kathy Hann established bounds on the average number of normals through a point in a convex bodyK, in the cases whereK is either a polytope or sufficiently smooth. In addition, an Euler-type theorem was obtained for these particular classes of convex bodies. In the present work we show that all these statements are true for an arbitrary convex bodyK. For this purpose measure geometric tools and a general approximation technique will be essential.