中国数学规划新近进展及展望.PDF

中国数学规划新近进展及展望 摘要 数学规划又称数学优化，它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。在本 章中，首先我们简单地介绍数学规划的历史、应用及其主要研究方向；然后我们概述数学规 划的发展现状和在中国的发展情况。最后我们将按照数学规划的以下七个主要方向：（1）线 性和非线性规划，（2 ）锥和鲁棒优化，（3 ）变分不等式和互补问题，（4 ）多目标优化与向量 优化，（5 ）整数规划，（6 ）组合优化，（7 ）张量与多项式优化，分别介绍其背景和应用领域， 研究现状，未来发展趋势和主要研究问题。 Recent Development and Future Prospect of Mathematical Programming in China Mathematical programming or mathematical optimization is an important branch of operations research that studies the problem of minimizing or maximizing a real function of real or integer variables, subject to constraints on the variables. It is one of widely used modeling tools and methodologies in operations research and management science and has numerous applications in engineering, economics and finance. In this chapter, we first give a brief introduction of mathematical programming problems, applications, history and main research areas. We then review the state-of-the-art of mathematical programming study with an overview of the development of mathematical programming in China. Research perspectives of mathematical programming is also presented. The main parts of the chapter devote to the following seven research areas of mathematical programming: (1) Linear and nonlinear programming; (2) Conic and robust optimization; (3) Variational inequality and complementarity problem; (4) Multi-objective optimization and vector optimization; (5) Integer programming; (6) Combinatorial optimization; (7) Tensor and polynomial optimization. In each of the above research areas, we introduce background and applications of the problems, the state-of-the-art o