Queue inference from periodic reporting data

We introduce new algorithms for queue inference problems involving periodic reporting data. We assume that in each of several consecutive time periods, new customer arrivals follow a Poisson process with period-specific arrival rates, but only the number of departing customers during each period is observed. Rather than exploiting relations with order statistics, our algorithms rely on direct recursions involving Poisson probabilities. We illustrate with several examples, some of which are not typically viewed as queueing applications.     

Busy period distributions in the steady state single-server queue and their applications

A method of generating probability distributions of the number of customers served in a busy period of a steady state single-server queueing system with univariate and multivariate inputs is described. A few moments in terms of the moments of the input distributions are derived. Some applications of the busy period distributions to such areas as branching processes, traffic flows, first passage problems, ballot theorems and waiting time distributions are briefly mentioned.     

Conjugate Gradient Method for Rank Deficient Saddle Point Problems

We propose an alternative iterative method to solve rank deficient problems arising in many real applications such as the finite element approximation to the Stokes equation and computational genetics. Our main contribution is to transform the rank deficient problem into a smaller full rank problem, with structure as sparse as possible. The new system improves the condition number greatly. Numerical experiments suggest that the new iterative method works very well for large sparse rank deficient saddle point problems.     

Quasidifferentiability in nonsmooth,nonconvex mechanics

Nonconvex and nonsmooth optimization problems arise in advanced engineering analysis and structural analysis applications. In fact the set of inequality and complementarity relations that describe the structural analysis problem are generated as optimality conditions by the quasidifferential potential energy optimization problem. Thus new kind of variational expressions arise for these problems, which generalize the classical variational equations of smooth mechanics, the variational inequalities of convex, nonsmooth mechanics and give a solid, computationally efficient explication of hemivariational inequalities of nonconvex, nonsmooth mechanics. Moreover quasidifferential calculus and optimization software make this approach applicable for a large number of problems. The connection of quasidifferential optimization and nonsmooth, nonconvex mechanics is discussed in this paper. A number of representative examples from elastostatic analysis applications are treated in details. Numerical examples illustrate the theory.     

An approximation method to model multivariate interpolation problems: Indexing HDMR

Dealing with univariate or bivariate data sets instead of a multivariate data set is an important concern in interpolation problems and computer-based applications. This paper presents a new data partitioning method that partitions the given multivariate data set into univariate and bivariate data sets and constructs an approximate analytical structure that interpolates function values at arbitrarily distributed points of the given grid. A number of numerical implementations are also given to show the performance of this new method.     

Sequential Quadratic Programming Methods for Large-Scale Problems

Sequential quadratic (SQP) programming methodsare the method of choice when solving small or medium-sized problems. Sincethey are complex methods they are difficult (but not impossible) to adapt tosolve large-scale problems. We start by discussing the difficulties that needto be addressed and then describe some general ideas that may be used toresolve these difficulties. A number of SQP codes have been written to solve specific applications and there is a general purposed SQP code called SNOPT,which is intended for general applications of a particular type. These aredescribed briefly together with the ideas on which they are based. Finally wediscuss new work on developing SQP methods using explicit second derivatives.     

On a generalization of Kronecker’s limit formula

In this paper, we first generalize the Kronecker limit formula for a class of Epstein zeta functions using new approximation formulas. This enables us to derive some applications to the class number of quadratic imaginary number fields K and the period ratios of elliptic curves with complex multiplication.     

Inverse Problems in Darboux’ Theory of Integrability

The Darboux theory of integrability for planar polynomial differential equations is a classical field, with connections to Lie symmetries, differential algebra and other areas of mathematics. In the present paper we introduce the concepts, problems and inverse problems, and we outline some recent results on inverse problems. We also prove a new result, viz. a general finiteness theorem for the case of prescribed integrating factors. A?number of relevant examples and applications is included.     

Quasi-equilibrium problems with applications to infinite optimization and constrained games in general topological spaces

By applying a new Fan-Browder type fixed point point theorem due to author, an existence theorem of quasi-equilibrium problem is proved in general topological space. As applications, some existence theorems of solutions for noncompact infinite optimization problems and noncompact constrained game problems are obtained in general topological spaces. These theorems improve and generalize a number of important results in recent literature.     

Explicit Runge–Kutta pairs appropriate for engineering applications

Runge–Kutta (RK) pairs of orders seven and five with minimal phase lag are derived for the numerical approximation of ordinary differential equations with engineering applications. For a class of initial value problems, whose solution is known to be described by free oscillations or free oscillations of high frequency with forced oscillations of low frequency superimposed, the new pair seem to offer clear advantages with respect to older pairs. The new pair is much more efficient than methods using the same number of stages, when applied in some problems of the plate deflection, the wave equation or vibratory systems.     

一些高振荡积分、高振荡积分方程的高性能计算

电磁、声波散射问题的研究涉及一类数学物理问题, 此类问题具有深刻的理论价值和重要的应用背景, 亟待解决. 高振荡微分、积分方程是刻画这些问题的重要的数学模型, 其数值计算存在许多挑战性研究课题. 本文从积分方程解法角度出发, 综述了求解这类高振荡问题的一些最新进展, 特别是针对广义Fourier 变换、Bessel 变换的高效算法、高振荡核Volterra 积分方程的数值解法作了详细介绍. 这些数值方法共有特点是振荡频率越高算法精度愈高, 且可望为电磁计算的研究提供一些新的高效算法.     

One-Dimensional Cutting Stock Problem with a Given Number of Setups: A Hybrid Approach of Metaheuristics and Linear Programming

One-dimensional cutting stock problem (1D-CSP) is one of the representative combinatorial optimization problems, which arises in many industrial applications. Since the setup costs for switching different cutting patterns become more dominant in recent cutting industry, we consider a variant of 1D-CSP, called the pattern restricted problem (PRP), to minimize the number of stock rolls while constraining the number of different cutting patterns within a bound given by users. For this problem, we propose a local search algorithm that alternately uses two types of local search processes with the 1-add neighborhood and the shift neighborhood, respectively. To improve the performance of local search, we incorporate it with linear programming (LP) techniques, to reduce the number of solutions in each neighborhood. A sensitivity analysis technique is introduced to solve a large number of associated LP problems quickly. Through computational experiments, we observe that the new algorithm obtains solutions of better quality than those obtained by other existing approaches.     

Lyapunov inequalities for the periodic boundary value problem at higher eigenvalues

This paper is devoted to provide some new results on Lyapunov type inequalities for the periodic boundary value problem at higher eigenvalues. Our main result is derived from a detailed analysis on the number and distribution of zeros of nontrivial solutions and their first derivatives, together with the study of some special minimization problems. This allows to obtain the optimal constants. Our applications include the Hill's equation where we give some new conditions on its stability properties and also the study of periodic and nonlinear problems at resonance where we show some new conditions which allow to prove the existence and uniqueness of solutions.     

Stability analysis of delay differential equation models of HIV-1 therapy for fighting a virus with another virus

Considering two kinds of delays accounting, respectively, for (i) a latent period between the time target cells are contacted by the virus particles and the time the virions enter the cells and (ii) a virus production period for new virions to be produced within and released from the infected cells, we develop and analyze a mathematical model for HIV-1 therapy by fighting a virus with another virus. For the different values of the basic reproduction number and the second basic reproduction number, we investigate the stability of the infection-free equilibrium, the single-infection equilibrium and the double-infection equilibrium. We conclude that increasing delays will decrease the values of the basic reproduction number and the second basic reproduction number. Our results have potential applications in HIV-1 therapy. The approach we use here is a combination of analysis of characteristic equations, Fluctuation Lemma and Lyapunov function.     

Applications of decision networks

In the last three decades, the ‘Principle of Optimality’ has been applied to a number of problems of multi-stage character. In this paper, we wish to review some of the work done by different authors using the principle.The current emphasis is on Decision Networks, which incorporates the basic principle and yet reveals the graphical structure of the problem similar to PERT/CPM charts. Actual applications of this technique are discussed in detail. It is expected that many new and real-life problems will be amenable to this technique.     

The Trended Inventory Lot Sizing Problem with Shortages Under a New Replenishment Policy

In the past few years, considerable attention has been given to the inventory lot sizing problem with trended demand over a fixed horizon. The traditional replenishment policy is to avoid shortages in the last cycle. Each of the remaining cycles starts with a replenishment and inventory is held for a certain period which is followed by a period of shortages. A new replenishment policy is to start each cycle with shortages and after a period of shortages a replenishment should be made. In this paper, we show that this new type of replenishment policy is superior to the traditional one. We further propose four heuristic procedures that follow the new replenishment policy. These are the constant demand approximation method, the equal cycle length heuristic, the extended Silver approach, and the extended least cost solution procedure. We also examine the cost and computation time performances of these heuristic procedures through an empirical study. The number of test problems solved to optimality, average and maximum cost deviation from optimum were used as measures of cost performance. The results of the 10 000 test problems reveal that the extended least cost approach is most cost effective.     

Controllable Systems with Impacts

Control problems in dynamic systems allowing pulse forces due to the presence of unilateral active constraints are considered. This paper gives an analysis of controls used during impact interactions. A new kind of pulse controls arising at the time of impact is introduced. These problems are typical for the so-called vibroimpact systems with numerous applications, whose number is constantly increasing in recent years. The goal of the paper is to describe the phase of a controllable impact and to determine the constraint reactions.     

A Benders decomposition method for discretely-constrained mathematical programs with equilibrium constraints

We present a new methodology to solve discretely-constrained mathematical programs with equilibrium constraints (DC-MPECs). Typically these problems include an upper planning-level optimization with some discrete decision variables (eg, build/don’t build) as well as a lower operations-level problem often described by an optimization or nonlinear complementarity problem. This lower-level problem may also include some discrete variables. MPECs are very challenging problems to solve and the inclusion of integrality constraints makes this class of problems even more computationally difficult. We develop a new variant of the Benders algorithm combined with a heuristic procedure that decomposes the domain of the upper-level discrete variables to solve the resulting DC-MPECs. We provide convergence theory as well as a number of numerical examples, some derived from energy applications, to validate the new method. It should be noted that the convergence theory applies if the heuristic procedure correctly identifies a decomposition of the domain so that the lower-level problem's optimal value function is convex. This is challenging but our numerical results are positive.     

Facets for node packing

The Node Packing Problem is an extremely important problem given that it comprises the underlying structure of numerous optimization problems either directly or indirectly. This paper presents a constraint approach which produces new facets for the Node Packing Problem. A number of different problem applications are solved incorporating this new constraint approach using a commercial software package on a personal computer demonstrating the effectiveness of the underlying facets in practice. The new facet structures provide a means for addressing general dispersion and separation requirements using mathematical programming.     

组合拓扑方法在组合学和图论中的应用

本文介绍组合拓扑方法在图论和组合学中的应用，探索一些新的离散问题和连续问题的关系，介绍目前有关这方面的新结果及发展动向。本文主要介绍同调理论在图论中的应用，与图有关的复形及性质，不动点定理在离散问题中的应用等。文中提出了一些新结果及可供研究的新问题。