中国数学规划学科发展概述

数学规划又称数学优化, 是运筹学的一个重要分支. 它主要研究在一定约束条件下, 如何求一个实数或者整数变量的实函数的最大值或者最小值. 它是运筹学和管理科学中最常用的一种建模工具和求解问题的方法, 在工程、经济和金融等领域有非常广泛的应用. 首先简单介绍数学规划的发展历史、应用领域及其主要研究方向; 然后简述数学规划的发展现状和在中国的发展进程; 最后, 讨论数学规划若干研究前沿问题与研究展望.     

Advances in Studies and Applications of Centroidal Voronoi Tessellations

<正>Centroidal Voronoi tessellations(CVTs) have become a useful tool in many applications ranging from geometric modeling,image and data analysis,and numerical partial differential equations,to problems in physics,astrophysics,chemistry,and biology. In this paper,we briefly review the CVT concept and a few of its generalizations and well-known properties.We then present an overview of recent advances in both mathematical and computational studies and in practical applications of CVTs.Whenever possible,we point out some outstanding issues that still need investigating.     

Operations research and data mining

With the rapid growth of databases in many modern enterprises data mining has become an increasingly important approach for data analysis. The operations research community has contributed significantly to this field, especially through the formulation and solution of numerous data mining problems as optimization problems, and several operations research applications can also be addressed using data mining methods. This paper provides a survey of the intersection of operations research and data mining. The primary goals of the paper are to illustrate the range of interactions between the two fields, present some detailed examples of important research work, and provide comprehensive references to other important work in the area. The paper thus looks at both the different optimization methods that can be used for data mining, as well as the data mining process itself and how operations research methods can be used in almost every step of this process. Promising directions for future research are also identified throughout the paper. Finally, the paper looks at some applications related to the area of management of electronic services, namely customer relationship management and personalization.     

数学建模中的数据挖掘方法

结合我校数模培训和参赛的经验，介绍了数据挖掘中的多元回归分析，主成分分析、人工神经网络等方法在建模中的一些成功应用。     

50 years of data mining and OR: upcoming trends and challenges

Data mining involves extracting interesting patterns from data and can be found at the heart of operational research (OR), as its aim is to create and enhance decision support systems. Even in the early days, some data mining approaches relied on traditional OR methods such as linear programming and forecasting, and modern data mining methods are based on a wide variety of OR methods including linear and quadratic optimization, genetic algorithms and concepts based on artificial ant colonies. The use of data mining has rapidly become widespread, with applications in domains ranging from credit risk, marketing, and fraud detection to counter-terrorism. In all of these, data mining is increasingly playing a key role in decision making. Nonetheless, many challenges still need to be tackled, ranging from data quality issues to the problem of how to include domain experts' knowledge, or how to monitor model performance. In this paper, we outline a series of upcoming trends and challenges for data mining and its role within OR.     

Graph spectra in Computer Science

In this paper, we shall give a survey of applications of the theory of graph spectra to Computer Science. Eigenvalues and eigenvectors of several graph matrices appear in numerous papers on various subjects relevant to information and communication technologies. In particular, we survey applications in modeling and searching Internet, in computer vision, data mining, multiprocessor systems, statistical databases, and in several other areas. Some related new mathematical results are included together with several comments on perspectives for future research. In particular, we claim that balanced subdivisions of cubic graphs are good models for virus resistent computer networks and point out some advantages in using integral graphs as multiprocessor interconnection networks.     

On optimization,dynamics and uncertainty: A tutorial for gene-environment networks

An emerging research area in computational biology and biotechnology is devoted to mathematical modeling and prediction of gene-expression patterns; to fully understand its foundations requires a mathematical study. This paper surveys and mathematically expands recent advances in modeling and prediction by rigorously introducing the environment and aspects of errors and uncertainty into the genetic context within the framework of matrix and interval arithmetic. Given the data from DNA microarray experiments and environmental measurements we extract nonlinear ordinary differential equations which contain parameters that are to be determined. This is done by a generalized Chebychev approximation and generalized semi-infinite optimization. Then, time-discretized dynamical systems are studied. By a combinatorial algorithm which constructs and follows polyhedra sequences, the region of parametric stability is detected. Finally, we analyze the topological landscape of gene-environment networks in terms of structural stability. This pioneering work is practically motivated and theoretically elaborated; it is directed towards contributing to applications concerning better health care, progress in medicine, a better education and more healthy living conditions.     

A remark on multiobjective stochastic optimization via strongly convex functions

Many economic and financial applications lead (from the mathematical point of view) to deterministic optimization problems depending on a probability measure. These problems can be static (one stage), dynamic with finite (multistage) or infinite horizon, single objective or multiobjective. We focus on one-stage case in multiobjective setting. Evidently, well known results from the deterministic optimization theory can be employed in the case when the “underlying” probability measure is completely known. The assumption of a complete knowledge of the probability measure is fulfilled very seldom. Consequently, we have mostly to analyze the mathematical models on the data base to obtain a stochastic estimate of the corresponding “theoretical” characteristics. However, the investigation of these estimates has been done mostly in one-objective case. In this paper we focus on the investigation of the relationship between “characteristics” obtained on the base of complete knowledge of the probability measure and estimates obtained on the (above mentioned) data base, mostly in the multiobjective case. Consequently we obtain also the relationship between analysis (based on the data) of the economic process characteristics and “real” economic process. To this end the results of the deterministic multiobjective optimization theory and the results obtained for stochastic one objective problems will be employed.     

The variable projection algorithm in time-resolved spectroscopy,microscopy and mass spectrometry applications

Nonlinear least squares optimization problems in which the parameters can be partitioned into two sets such that optimal estimates of parameters in one set are easy to solve for given fixed values of the parameters in the other set are common in practice. Particularly ubiquitous are data fitting problems in which the model function is a linear combination of nonlinear functions, which may be addressed with the variable projection algorithm due to Golub and Pereyra. In this paper we review variable projection, with special emphasis on its application to matrix data. The generalization of the algorithm to separable problems in which the linear coefficients of the nonlinear functions are subject to constraints is also discussed. Variable projection has been instrumental for model-based data analysis in multi-way spectroscopy, time-resolved microscopy and gas or liquid chromatography mass spectrometry, and we give an overview of applications in these domains, illustrated by brief case studies.     

向量优化及其若干进展

在一定的约束条件下极小化或极大化向量值函数,这就是向量优化. 向量优化是数学规划学科中的重要分支学科,是具有重要应用价值的、新兴的和多学科交叉的研究领域. 自1950年以来,已经逐步形成较完整的理论体系,算法研究也有一定的进展,应用日渐广泛. 简述了它的发展历程、主要特征、基本理论和方法,综述了国内学者近几年来在若干领域的发展状况和主要代表性成果,展望了向量优化学科未来的发展方向.     

Reflection groups in analysis and applications

This is an overview of the use of reflection groups in analysis, applications in algebra, mathematical physics, and probability.      

金属矿山企业生产计划优化方法

针对金属矿山企业的单位开采与运输成本大、优化求解结果偏差大问题, 首先, 依据金属矿山企业编制开采计划的基本原则, 以矿石开采与运输成本最小化为优化目标, 利用整数规划方法, 构建了金属矿山企业生产计划数学模型, 其次, 为了精准快速求解金属矿山企业生产计划模型, 提出了改进的量子粒子群优化算法, 采用进化速度和聚集度因子对算法中的惯性权重进行动态调整, 并设计了双层可行域搜索策略, 提高了算法的局部和全局搜索能力。最后, 以某大型金属矿山企业采运生产作业为案例, 通过与矿山实际生产指标、非线性规划结果以及粒子群优化结果进行比较分析。结果表明:在相同经济指标和参数环境下, 本文算法优于其它两种优化方法, 且每吨矿石的开采和运输成本减少了0.05元左右, 降低了金属矿山企业的开采运输成本, 提高了企业的整体经济效益。     

Characterizations of linear suboptimality for mathematical programs with equilibrium constraints

The paper is devoted to the study of a new notion of linear suboptimality in constrained mathematical programming. This concept is different from conventional notions of solutions to optimization-related problems, while seems to be natural and significant from the viewpoint of modern variational analysis and applications. In contrast to standard notions, it admits complete characterizations via appropriate constructions of generalized differentiation in nonconvex settings. In this paper we mainly focus on various classes of mathematical programs with equilibrium constraints (MPECs), whose principal role has been well recognized in optimization theory and its applications. Based on robust generalized differential calculus, we derive new results giving pointwise necessary and sufficient conditions for linear suboptimality in general MPECs and its important specifications involving variational and quasivariational inequalities, implicit complementarity problems, etc. Research was partially supported by the National Science Foundation under grant DMS-0304989 and by the Australian Research Council under grant DP-0451168.     

Mathematical modelling and parameter optimization of pulsating heat pipes

Proper heat transfer management is important to key electronic components in microelectronic applications. Pulsating heat pipes (PHP) can be an efficient solution to such heat transfer problems. However, mathematical modelling of a PHP system is still very challenging, due to the complexity and multiphysics nature of the system. In this work, we present a simplified, two-phase heat transfer model, and our analysis shows that it can make good predictions about startup characteristics. Furthermore, by considering parameter estimation as a nonlinear constrained optimization problem, we have used the firefly algorithm to find parameter estimates efficiently. We have also demonstrated that it is possible to obtain good estimates of key parameters using very limited experimental data.     

Combining metaheuristics with mathematical programming, constraint programming and machine learning

During the last years, interest on hybrid metaheuristics has risen considerably in the field of optimization and machine learning. The best results found for many optimization problems in science and industry are obtained by hybrid optimization algorithms. Combinations of optimization tools such as metaheuristics, mathematical programming, constraint programming and machine learning, have provided very efficient optimization algorithms. Four different types of combinations are considered in this paper: (i) Combining metaheuristics with complementary metaheuristics. (ii) Combining metaheuristics with exact methods from mathematical programming approaches which are mostly used in the operations research community. (iii) Combining metaheuristics with constraint programming approaches developed in the artificial intelligence community. (iv) Combining metaheuristics with machine learning and data mining techniques.     

Blind source separation with nonlinear autocorrelation and non-Gaussianity

Blind source separation (BSS) is a problem that is often encountered in many applications, such as biomedical signal processing and analysis, speech and image processing, wireless telecommunication systems, data mining, sonar, radar enhancement, etc. One often solves the BSS problem by using the statistical properties of original sources, e.g., non-Gaussianity or time-structure information. Nevertheless, real-life mixtures are likely to contain both non-Gaussianity and time-structure information sources, rendering the algorithms using only one statistical property fail. In this paper, we address the BSS problem when source signals have non-Gaussianity and temporal structure with nonlinear autocorrelation. Based on the two statistical characteristics of sources, we develop an objective function. Maximizing the objective function, we propose a gradient ascent source separation algorithm. Furthermore, We give some mathematical properties for the algorithm. Computer simulations for sources with square temporal autocorrelation and non-Gaussianity illustrate the efficiency of the proposed approach.     

Immersive Projection Technology for Visual Data Mining

The PlatoCAVE, the MiniCAVE, and the C2 are immersive stereoscopic projectionbased virtual reality environments oriented toward group interactions. As such they are particularly suited to collaborative efforts in data analysis and visual data mining. In this article, we provide an overview of virtual reality in general, including immersive projection technology, and the use of stereoscopic displays for data visualization. We discuss design considerations for the construction of these immersive environments including one-wall versus four-wall implementations, augmented reality, stereoscopic placement, head tracking, the use of LCD devices, polarized light stereo, voice control, and image synchronization.     

Super parametric convex model and its application for non-probabilistic reliability-based design optimization

In this study, we attempt to propose a new super parametric convex model by giving the mathematical definition, in which an effective minimum volume method is constructed to give a reasonable enveloping of limited experimental samples by selecting a proper super parameter. Two novel reliability calculation algorithms, including nominal value method and advanced nominal value method, are proposed to evaluate the non-probabilistic reliability index. To investigate the influence of non-probabilistic convex model type on non-probabilistic reliability-based design optimization, an effective approach based on advanced nominal value method is further developed. Four examples, including two numerical examples and two engineering applications, are tested to demonstrate the superiority of the proposed non-probabilistic reliability analysis and optimization technique.     

A survey of some mathematical programming models in transportation

Transportation problems constitute a fertile domain for the application of mathematical programming models and nonlinear optimization techniques, distribution problems, entropy models, traffic assigment problems and many others are good examples of this assertion. This paper provides a summary overview of the main modeling approaches in transportation and the related optimization models, symmetric and asymmetric, and an overview on the state-of-the-art of the origindestination adjustment problems and the related bilevel optimization methods.     

Hahn-Banach extension theorems over the space of fuzzy elements

The conventional Hahn-Banach extension theorem over a vector space has been widely used to derive many important and interesting results in nonlinear analysis, vector optimization and mathematical economics. Although the space of fuzzy elements is not a real vector space, the Hahn-Banach extension theorems over the space of fuzzy elements and the nonstandard normed space of fuzzy elements are presented in this paper. The work also shows the possible applications of the fuzzy-valued problems to nonlinear analysis, vector optimization and mathematical economics.