CAREER: Linear Matrix Inequality Representations in Optimization

职业：优化中的线性矩阵不等式表示

• 批准号: 0844775
• 负责人: Jiawang Nie
• 金额: $ 50.04万
• 依托单位: University of California-San Diego
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-09-01 至 2015-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0844775&HistoricalAwards=false
• 关键词: CAREER; Linear; Matrix; Inequality; Representations

This proposal investigates the linear matrix inequality representations of convex sets and their applications in optimization problems. The work involves different kinds of mathematical tools like algebraic geometry, convex analysis, differential geometry, numerical analysis, optimization theory, and real algebra. The investigator not only studies the fundamental mathematics on the scope and depth of linear matrix inequality representability, but also work on designing new algorithms and software solving hard optimization problems.The following five main topics will be focused in this project: linear matrix inequality representations of rigid convex sets, semidefinite programming representations of convex semialgebraic sets, second order cone programming representations of convex semialgebraic sets, semidefinite programming representations of nonnegative multivariate polynomials, and linear matrix inequality methods for solving nonconvex optimization problems and polynomial systems. A basic problem of science and engineering is finding a global minimum of a function of many variables. As a metaphor one might think of a complicated terrain of mountains and valleys which stretches for hundreds of miles and one must find the lowest point in the lowest valley. The difficulty is that one can not see the map and one only knows a mathematical formula for the terrain and in most applications (like electronics, networks, biochemistry) there are many variables instead of three. Many algorithms will find the lowest point of a particular valley but none are known which effectively find the lowest valley itself. This NSF research is to develop global optimization algorithms for various situations. One is the class of problems where the data is given by polynomials. Another is to determine and parameterize convex problems very efficiently; in convex situations one has only one valley. These pursuits require integration of techniques from numerical mathematics, real and complex algebraic geometry, convex analysis, differential geometry, numerical analysis, and optimization theory, a wide range of mathematics. Jiawang Nie has personal experience with several areas of applications including sensor networks and systems control and this informs his mathematics and techniques. Other important features of this proposal are integrating research and education, developing new mathematical courses, training undergraduate and graduate students on using the latest mathematical tools, advising postdoctoral scholars on how to create novel research results.

该建议研究了凸集的线性矩阵不等式表示及其在优化问题中的应用。这项工作涉及多种数学工具，例如代数几何，凸分析，差异几何，数值分析，优化理论和真实代数。 The investigator not only studies the fundamental mathematics on the scope and depth of linear matrix inequality representability, but also work on designing new algorithms and software solving hard optimization problems.The following five main topics will be focused in this project: linear matrix inequality representations of rigid convex sets, semidefinite programming representations of convex semialgebraic sets, second order cone programming representations of凸半缘式集合，非负多元多项式的半标准编程表示以及用于解决非convex优化问题和多项式系统的线性矩阵不等式方法。 科学和工程学的一个基本问题是找到许多变量的函数的全球最低限度。作为一个隐喻，人们可能会想到一个复杂的山脉和山谷地形，延伸了数百英里，必须找到最低山谷中的最低点。困难是一个人看不到地图，一个人只知道地形的数学公式，在大多数应用程序（例如电子，网络，生物化学）中，有很多变量而不是三个变量。许多算法会发现特定山谷的最低点，但没有人有效地找到最低的山谷本身。这项NSF研究是为各种情况开发全球优化算法。一个是多项式给出数据的问题类别。另一个是非常有效地确定和参数化凸问题。在凸状态下，一个人只有一个山谷。这些追求需要从数值数学，真实和复杂的代数几何形状，凸分析，差异几何形状，数值分析和优化理论（广泛的数学范围）中整合技术。 Jiawang Nie在包括传感器网络和系统控制在内的几个应用领域具有个人经验，这为他的数学和技术提供了信息。该建议的其他重要特征是整合研究和教育，开发新的数学课程，培训本科生和研究生使用最新的数学工具，并向博士后学者提供有关如何创建新颖研究结果的建议。