On dual vector equilibrium problems

In this work, we consider the dual vector equilibrium problems in the topological vector spaces setting for a moving cone. We investigate the relationship between solutions of the vector equilibrium problems and those for their perturbations. Our result may be viewed as a refinement and improvement of the paper [L. Huang, Existence of Solutions on Weak Vector Equilibrium Problems, vol. 65, 2006, pp. 795–801.].     

The influence of perturbations on stability in terms of two metrics

The problem of stability under perturbations in terms of two metrics is investigated. Problems of the stability of the equilibrium position of a mechanical system with variable mass under perturbations are solved.     

Limit distributions of random symmetrical polynomials

Conditions are studied which should be imposed on the coefficients of a homogeneous random polynomial of the fourth degree to provide its convergence to some nondegenerate random variable. Proceedings of the XVII Seminar on Stability Problems for Stochastic Models, Kazan, Russia, 1995, Part II.     

A reduction principle for asymptotically triangular differential systems

Problems of the stability of non-linear non-autonomous systems of differential equations with a special class of asymptotically vanishing perturbations are considered. The problem of reducing a problem on the stability of the equilibrium of a perturbed system to a problem on stability with respect to a non-linear approximation system which has a triangular form is solved. Applications of the results of the investigations to mechanical systems with a variable mass and time-varying equations of the constraints are presented.     

Models and heuristic algorithms for a weighted vertex coloring problem

We consider a weighted version of the well-known Vertex Coloring Problem (VCP) in which each vertex i of a graph G has associated a positive weight w _i . Like in VCP, one is required to assign a color to each vertex in such a way that colors on adjacent vertices are different, and the objective is to minimize the sum of the costs of the colors used. While in VCP the cost of each color is equal to one, in the Weighted Vertex Coloring Problem (WVCP) the cost of each color depends on the weights of the vertices assigned to that color, and it equals the maximum of these weights. WVCP is known to be NP-hard and arises in practical scheduling applications, where it is also known as Scheduling on a Batch Machine with Job Compatibilities. We propose three alternative Integer Linear Programming (ILP) formulations for WVCP: one is used to derive, dropping integrality requirement for the variables, a tight lower bound on the solution value, while a second one is used to derive a 2-phase heuristic algorithm, also embedding fast refinement procedures aimed at improving the quality of the solutions found. Computational results on a large set of instances from the literature are reported.     

幂向量,复合向量数及其函数理论

本文提出向量为其幂向量和向量幂级数．向量幂级数由一实数和某一向量联合组成的“复合向量数”及其函数有重要涵义．这数也有运算法则．从复合向量数的函数理论分析知其函数有导数和解析函数的必要和充分条件．这些条件构成了“双曲型”方程的特征以及函数的积分性质等．     

On fisher's information contained in the sample mean

Within the framework of a parametric model, it is assumed that a number of first moments of the observed random variable are known. It is shown how fast the Fisher information contained in the sample mean decreases as the sample size tends to infinity. Proceedings of the Seminar on Stability Problems for Stochastic Models, Hajdúszoboszló, Hungary, 1997, Part II.     

Reducing the number of variables in Integer and Linear Programming Problems

Computational difficulties in solving the Integer Programming Problems (IPP) are caused to a considerable degree by the number of variables. If the number of variables is small, then even NP-complete problems usually can be solved with a reasonable expenditure of effort.A procedure is developed for the analysis of large scale IPP with the aim of reducing the number of variables prior to starting the solution method. The procedure is based on comparing pairs of columns of the constraint matrix of the IPP. If a pair of columns thus compared meets certain conditions, then the IPP has an optimal solution, in which a variable corresponding to one of the columns in the pair is equal to zero. Corresponding theorems for Knapsack and Multidimensional Knapsack problems and for general IPP are presented. The procedure is extended to Linear and Mixed Integer Programming Problems. The presented results of computational experiments illustrate the efficiency of the developed procedure.     

On characterizations of continuous distributions in terms of moments of order statistics when the sample size is random

We characterize continuous distribution functions F of a population when the sample size N is a random variable. Characterizing conditions in terms of moments of the kth order statistics are given, among other things, for the uniform, exponential, Pareto, logistic, and Weibull distributions. Proceedings of the Seminar on Stability Problems for Stochastic Models. Hajdúszoboszló, Hungary, 1997, Part II.     

Statistical inference for G/M/1 queueing system

In this paper, maximum likelihood estimates of the parameters are derived for the G/M/1 queueing model with variable arrival rate. A simulated numerical example is used to illustrate its application for estimating the parameter when the interarrival time distribution is exponential. Problems of hypothesis testing are also investigated.     

The central limit theorem without the condition of independence

Necessary and sufficient conditions are presented for sums of asymptotically independent random variables to converge to a normal random variable in the sense of total variation distance, uniform metric for characteristic functions. and mean metric of order q. Supported by the Russian Foundation for Fundamental Research (grant No. 96-01-01920). Proceedings of the Seminar on Stability Problems for Stochastic Models. Moscow. Russia. 1996. Part II.     

Nonuniform error bounds in asymptotic expansions for scale mixtures under mild moment conditions

Let F(x) be a distribution function of of a scale mixture X=SZ of a random variable Z with distribution G and scale factor S, which is a positive random variable independent of Z. Some nonuniform bounds are given for asymptotic expansions of F(x) around G(x)_ under mild moment conditions on the distribution of S. Some nonuniform bounds for the normal approximation to the Student t-distribution are given as examples. Supported by a Grant-in-Aid for the COE Research Program, The Minsitry of Education, Science, Sports, and Culture, Japan, and by the Russian Foundation for Fundamental Research (grant No. 96-01-01919). Proceedings of the Seminar on Stability Problems for Stochastic Models, Hajdúszoboszló, Hungary, 1997. Part III.     

A new subtour elimination constraint for the vehicle routing problem

Vehicle Routing Problems (VRP) are concerned with the delivery of a single commodity from a centralized depot to a number of specified customer locations with known demands. In this paper we consider the VRP characterized by: fixed or variable number of vehicles, common vehicle capacity, distance restrictions, and minimization of total distance travelled by all vehicles as the objective. We develop an exact algorithm based on a new subtour elimination constraint. The algorithm is implemented using the CPLEX package for solving the relaxed subproblems. Computational results on 1590 simulated problems and 10 literature problems (without distance restrictions) are reported and a comparative analysis is carried out.     

Hamilton-Jacobi theory over time scales and applications to linear-quadratic problems

In this paper we first derive the verification theorem for nonlinear optimal control problems over time scales. That is, we show that the value function is the only solution of the Hamilton-Jacobi equation, in which the minimum is attained at an optimal feedback controller. Applications to the linear-quadratic regulator problem (LQR problem) gives a feedback optimal controller form in terms of the solution of a generalized time scale Riccati equation, and that every optimal solution of the LQR problem must take that form. A connection of the newly obtained Riccati equation with the traditional one is established. Problems with shift in the state variable are also considered. As an important tool for the latter theory we obtain a new formula for the chain rule on time scales. Finally, the corresponding LQR problem with shift in the state variable is analyzed and the results are related to previous ones.     

A note on a characterization of the Poisson and uniform distributions

It is shown that if U, X are independent random variables, X≥0, U is uniformly distributed in (0,1), and X satisfies the equation UX+2∼U+X, then X−2 has the Poisson distribution with the parameter equal one. The above equation also characterizes the uniform distribution if X−2 is a Poisson random variable. Moreover, a multivariate generalization is given. Proceedings of the XVI Seminar on Stability Problems for Stochastic Models, Part I, Eger, Hungary, 1994.     

Constructive observability inequalities for weak generalized solutions of the wave equation with elastic restraint

Problems with one-sided boundary controls and a homogeneous Robin boundary condition set on the uncontrolled end are considered in the class of strong generalized solutions of the variable coefficient wave equation. In the adjoint class of weak generalized solutions of the dual problems with one-sided observations, new constructive observability inequalities are obtained that differ from previously known ones by an optimal threshold time. It is shown that, in the considered functional classes, the estimated constants degenerate as the time interval length approaches the threshold. Numerical illustrations are given showing that the stability of approximate solutions to control problems can be substantially enhanced by taking into account a priori information contained in the resulting observability inequalities.     

向量互补问题和极小元问题

本文不仅引入了非空集的有效下界的概念,而且引入了新的严格C-伪单调映射,在C-伪单调性的假设下,文内证明了向量互补问题（VCP）的可行集及其相关集合的有效下界和下有效点的存在性。     

结构VAR的有向非循环图模型

研究用图模型方法辨识结构向量自回归(VAR)模型,图中的结点表示不同时刻的随机变量,结点间的边表示其所表示的随机变量之间存在的因果相依关系.针对建立有向非循环图的问题,提出了一种基于回归分析的判断方法,用回归方程的回归平方和之差作为统计量,确定当前变量之间相依关系的方向.与R ea le的逐一判别法和A lessio的图搜索方法相比,文中提出的基于统计分析的方法简单易行,且可获得唯一的当前变量有向非循环图.最后以两组模拟序列为例,验证了所提出的方法是可行且有效的.     

Piece-wise linear approximation of functions of two variables

The goal of increasing computational efficiency is one of the fundamental challenges of both theoretical and applied research in mathematical modeling. The pursuit of this goal has lead to wide diversity of efforts to transform a specific mathematical problem into one that can be solved efficiently. Recent years have seen the emergence of highly efficient methods and software for solving Mixed Integer Programming Problems, such as those embodied in the packages CPLEX, MINTO, XPRESS-MP. The paper presents a method to develop a piece-wise linear approximation of an any desired accuracy to an arbitrary continuous function of two variables. The approximation generalizes the widely known model for approximating single variable functions, and significantly expands the set of nonlinear problems that can be efficiently solved by reducing them to Mixed Integer Programming Problems. By our development, any nonlinear programming problem, including non-convex ones, with an objective function (and/or constraints) that can be expressed as sums of component nonlinear functions of no more than two variables, can be efficiently approximated by a corresponding Mixed Integer Programming Problem.     

Chebyshev tau matrix method for Poisson-type equations in irregular domain

In this paper, a new meshless method, Chebyshev tau matrix method (CTMM) is researched. The matrix representations for the differentiation and multiplication of Chebyshev expansions make CTMM easy to implement. Problems with curve boundary can be efficiently treated by CTMM. Poisson-type problems, including standard Poisson problems, Helmholtz problems, problems with variable coefficients and nonlinear problems are computed. Some numerical experiments are implemented to verify the efficiency of CTMM, and numerical results are in good agreement with the analytical one. It appears that CTMM is very effective for Poisson-type problems in irregular domains.