Variable selection in model-based discriminant analysis

A general methodology for selecting predictors for Gaussian generative classification models is presented. The problem is regarded as a model selection problem. Three different roles for each possible predictor are considered: a variable can be a relevant classification predictor or not, and the irrelevant classification variables can be linearly dependent on a part of the relevant predictors or independent variables. This variable selection model was inspired by a previous work on variable selection in model-based clustering. A BIC-like model selection criterion is proposed. It is optimized through two embedded forward stepwise variable selection algorithms for classification and linear regression. The model identifiability and the consistency of the variable selection criterion are proved. Numerical experiments on simulated and real data sets illustrate the interest of this variable selection methodology. In particular, it is shown that this well ground variable selection model can be of great interest to improve the classification performance of the quadratic discriminant analysis in a high dimension context.     

Uncertain portfolio optimization problem under a minimax risk measure

Portfolio optimization problem is concerned with choosing an optimal portfolio strategy that can strike a balance between maximizing investment return and minimizing investment risk. In many cases, the return rate of risky asset is neither a random variable nor a fuzzy variable. Then, it can be described as an uncertain variable. But, the existing works on uncertain portfolio optimization problem fail to find an analytic solution of optimal portfolio strategy. In this paper, we define a new uncertain risk measure for the modeling of investment risk. Then, an uncertain portfolio optimization model is formulated. By introducing a new variable, we transform it into an equivalent bi-criteria optimization model. Then, we derive a method for the construction of the set of analytic Pareto optimal solutions. Finally, a numerical simulation is carried out to show the applicability of the proposed model and the convenience of finding the analytic solution.     

几种新的回归诊断图示模型及其医学应用

本文在变量选择问题的基础上，提出了一种新的图示模型──减变残差图。并给出它的两种推广形式：均值平移异常值检验图和部分影响诊断图。通过它们不但可以容易地考察一个变量在模型中的作用和检验异常值，而且可以诊断样本点对模型和变量的影响大小。     

An answer to Li's problem

In this paper, an answer is given to a problem proposed by K.C. Li. This problem comes from the variable transformation for fitting a regression model.     

Linear,Integer, Separable and Fuzzy Programming Problems: A Unified Approach towards Reformulation

For mathematical programming (MP) to have greater impact as a decision tool, MP software systems must offer suitable support in terms of model communication and modelling techniques. In this paper, modelling techniques that allow logical restrictions to be modelled in integer programming terms are described, and their implications discussed. In addition, it is illustrated that many classes of non-linearities which are not variable separable may be, after suitable algebraic manipulation, put in a variable separable form. The methods of reformulating the fuzzy linear programming problem as a max-min problem is also introduced. It is shown that analysis of bounds plays a key role in the following four important contexts: model reduction, reformulation of logical restrictions as 0-1 mixed integer programmes, reformulation of non-linear programmes as variable separable programmes and reformulation of fuzzy linear programmes. It is observed that, as well as incorporating an interface between the modeller and the optimizer, there is a need to make available to the modeller software facilities which support the model reformulation techniques described here.     

Determining the Optimum Stratum Boundaries Using Mathematical Programming

The method of choosing the best boundaries that make strata internally homogeneous, given some sample allocation, is known as optimum stratification. In order to make the strata internally homogeneous, the strata are constructed in such a way that the strata variances should be as small as possible for the characteristic under study. In this paper the problem of determining optimum strata boundaries (OSB) is discussed when strata are formed based on a single auxiliary variable with a varying measurement cost per units across strata. The auxiliary variable considered in the problem is a size variable that holds a common model for a whole population. The OSB are achieved effectively by assuming a suitable distribution of the auxiliary variable and creating strata by cutting the range of the distribution at optimum points. The problem of finding the OSB, which minimizes the variance of the estimated population mean under a weighted stratified balanced sampling, is formulated as a mathematical programming problem (MPP). Treating the formulated MPP as a multistage decision problem, a solution procedure using dynamic programming technique is developed. A numerical example using a hospital population data is presented to illustrate the computational details of the solution procedure. A software program coded in JAVA is written to carry out the computation. The distribution of the auxiliary variable in this example is considered to be continuous with an exponential density function.     

带时变生产成本的易变质经济批量模型的最优策略分析

考虑了具有时变生产成本的易变质产品经济批量模型．有限计划期内,单位生产成本、生产率以及需求率假定为时间的连续函数,生产固定成本则具有遗忘效应现象．当不允许缺货时,建立了以总成本最小为目标的混合整数优化模型并证明了此问题最优解的相关性质．对于此问题的特殊情形,将成本函数中的离散型变量松弛为连续型变量,通过分析其最优解的存在性及唯一性,求解了此最优解,将其作为初始值设计了求取一般情形最优解的有效算法．最后通过算例验证了理论结果的有效性．     

Optimal Control for non–isothermal crystallization of polymers

In this paper an optimal control problem for polymer crystallization is investigated. The crystallization is described by a non–isothermal Avrami–Kolmogorov model. The boundary temperature serves as control variable. The cost functional takes the spatial variation of the crystallinity and the final degree of crystallization into account. The resulting boundary control problem for a parabolic equation coupled with two ordinary differential equations is treated by an adjoint variable approach. A steepest descent algorithm is used to solve the problem numerically. Simulations illustrate the applicability and performance of the optimization algorithm. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Combinatorial Approach to Piecewise Linear Time Series Analysis

Over recent years, several nonlinear time series models have been proposed in the literature. One model that has found a large number of successful applications is the threshold autoregressive model (TAR). The TAR model is a piecewise linear process whose central idea is to change the parameters of a linear autoregressive model according to the value of an observable variable, called the threshold variable. If this variable is a lagged value of the time series, the model is called a self-exciting threshold autoregressive (SETAR) model. In this article, we propose a heuristic to estimate a more general SETAR model, where the thresholds are multivariate. We formulate the task of finding multivariate thresholds as a combinatorial optimization problem. We develop an algorithm based on a greedy randomized adaptive search procedure (GRASP) to solve the problem. GRASP is an iterative randomized sampling technique that has been shown to quickly produce good quality solutions for a wide variety of optimization problems. The proposed model performs well on both simulated and real data.     

An exact algorithm for the type-constrained and variable sized bin packing problem

In this paper, we introduce an additional constraint to the one-dimensional variable sized bin packing problem. Practically, some of items have to be packed separately in different bins due to their specific requirement. Therefore, these items are labelled as different types. The bins can be used to pack either any type of items if they are empty originally or the same type of items as what they already have. We model the problem as a type-constrained and variable sized bin packing problem (TVSBPP), and solve it via a branch and bound method. An efficient backtracking procedure is proposed to improve the efficiency of the algorithm.     

An integer programming model for hierarchical workforce scheduling problem

In this paper, an integer programming model for the hierarchical workforce problem under the compressed workweeks is developed. The model is based on the integer programming formulation developed by Billionnet [A. Billionnet, Integer programming to schedule a hierarchical workforce with variable demands, European Journal of Operational Research 114 (1999) 105–114] for the hierarchical workforce problem. In our model, workers can be assigned to alternative shifts in a day during the course of a week, whereas all workers are assigned to one shift type in Billionnet’s model. The main idea of this paper is to use compressed workweeks in order to save worker costs. This case is also suitable for the practice. The proposed model is illustrated on the Billionnet’s example problem and the obtained results are compared with the Billionnet’s model results.     

两级定位-路径问题模型及变邻域粒子群算法

为满足B2C电子商务中高效率、低成本配送需求,建立了两级定位-路径问题的三下标车流模型,提出了一种求解该问题的变邻域粒子群算法。该算法引入路径重连思想,将粒子群算法中粒子动态更新设计为当前解的邻域搜索、当前解与个体历史最优解之间的路径重连、当前解与种群历史最优解之间的路径重连;在此基础上,提出变邻域搜索策略,动态改变邻域结构以拓展搜索空间。实验结果表明,该算法能有效求解两级定位-路径问题。     

A probabilistic programming model for blending aggregates

The paper introduces the formulation of a probabilistic programming model to find the optimum mix proportion of aggregates to meet the specific grading requirement in order to minimize the cost which consists of the material cost and the expected penalty cost. The model is probabilistic since the gradation, which is the major parameter, is a random variable. A linear programming model is first formulated. Using the LP solution as initial value, a direct search technique is then employed to solve the problem. The model is expected to be applicable to any problem of aggregates blending. In this paper, however, the mixing aggregates of an asphalt mixing plant is exemplified to test the applicability of the model.     

基于可变航速的支线集装箱船舶调度优化模型与算法

随着航运市场的竞争不断加剧和集装箱船舶大型化的发展,越来越多的航运企业选择轴-辐式航运网络模式。支线船舶调度问题作为轴-辐式航运网络的重要组成部分受到研究者的高度关注。本文研究了可变航速和经济航速两种情境下的支线船舶调度问题,同时考虑枢纽港和喂给港的取送箱时间窗限制,以航运企业运营成本最小化为目标函数建立非线性混合整数规划模型。首先使用专业的规划求解器进行小规模算例的求解,验证了模型的准确性。同时运用改进的遗传算法对大规模支线船舶优化调度模型进行求解。为了提高求解效果,进一步设计了多智能体进化算法进行求解。数值结果表明,可变航速的运营成本低于经济航速的运营成本;在算法效率方面,改进遗传算法收敛速度较快,多智能体进化算法则可以提高求解精度。     

Distribution of the inter and intra inertia in conditional MCA

Conditional Multiple Correspondence Analysis (MCA), where a control variable plays the role of a partition model, allows us to decompose global inertia into between inertia and within inertia. The problem is to assess when the conditioning variable gives different results with respect to the unconditional analysis. In this paper, we study the asymptotic distribution function of these inertias, which can allow us to determine whether conditioning is significant. Some simulations were performed to corroborate the established results.     

Concave group methods for variable selection and estimation in high-dimensional varying coefficient models

The varying-coefficient model is flexible and powerful for modeling the dynamic changes of regression coefficients. We study the problem of variable selection and estimation in this model in the sparse, high-dimensional case. We develop a concave group selection approach for this problem using basis function expansion and study its theoretical and empirical properties. We also apply the group Lasso for variable selection and estimation in this model and study its properties. Under appropriate conditions, we show that the group least absolute shrinkage and selection operator (Lasso) selects a model whose dimension is comparable to the underlying model, regardless of the large number of unimportant variables. In order to improve the selection results, we show that the group minimax concave penalty (MCP) has the oracle selection property in the sense that it correctly selects important variables with probability converging to one under suitable conditions. By comparison, the group Lasso does not have the oracle selection property. In the simulation parts, we apply the group Lasso and the group MCP. At the same time, the two approaches are evaluated using simulation and demonstrated on a data example.     

Control of Crystallization Processes

In this paper an optimal control problem for polymer crystallization is investigated. The crystallization is described by a non–isothermal Avrami–Kolmogorov model. The boundary temperature serves as control variable. The cost functional takes the spatial variation of the crystallinity and the final degree of crystallization into account. The resulting boundary control problem for a parabolic equation coupled with two ordinary differential equations is treated by an adjoint variable approach. Numerical simulations are carried out for different scenarios to illustrate the applicability and performance of the optimization algorithm. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Reconstruction of controls in exponentially stable linear systems subjected to small perturbations

The problem of the dynamical reconstruction of the variable input of an exponentially stable linear system subjected to small non-linear perturbations is considered. In the case of inaccurate observations of its phase trajectory, an algorithm for solving this problem is given, based on the method of control with a model. The algorithm is stable to data interference and computation errors. General constructions are illustrated by an example in which the problem of reconstructing the input of an oscillatory section is discussed.     

Vibrations of the Beam with Variable Cross‐Section Caused by Moving Stochastic Load

We present the problem of vibrations of a beam with variable geometry which are caused by stochastic moving load. The approach is based on concepts of the tolerance averaged model [4]. In this way we formulate the averaged equations of the structured beam which describe the length‐scale effect. Using the averaged equations we obtain the probabilistic characteristics of the beam with periodically variable geometry.     

Some mathematical properties of radiating gas model obtained with a variable Eddington factor

Some mathematical properties of the model for a radiating gas obtained with a variable Eddington factor are studied.In particular we discuss the problem of symmetrization and the derivation of so-called a priori estimates for the quasilinear equations describing the model.