CAREER: Linear Matrix Inequality Representations in Optimization

本文研究凸集的线性矩阵不等式表示及其在最优化问题中的应用。这项工作涉及不同类型的数学工具，如代数几何、凸分析、微分几何、数值分析、最优化理论和实代数。研究人员不仅研究了线性矩阵不等式表示的广度和深度的基本数学问题，而且还致力于设计新的算法和软件来解决困难的优化问题。本项目将重点研究以下五个主要问题：刚性凸集的线性矩阵不等式表示，凸半代数集的半定规划表示，凸半代数集的二阶锥规划表示，非负多元多项式的半定规划表示，以及求解非凸优化问题和多项式系统的线性矩阵不等式方法。科学和工程中的一个基本问题是寻找一个多元函数的全局极小值。作为一个比喻，你可能会想到一个复杂的山脉和山谷的地形，绵延数百英里，你必须在最低的山谷中找到最低点。困难在于，人们看不到地图，只知道地形的数学公式，在大多数应用中(如电子、网络、生物化学)，有许多变量而不是三个变量。许多算法会找到特定山谷的最低点，但没有一种算法能有效地找到最低山谷本身。NSF的这项研究是为了开发适用于各种情况的全局优化算法。一类是数据由多项式给出的问题。另一种是非常有效地确定和参数化凸问题；在凸的情况下，一个人只有一个谷。这些追求需要整合来自数值数学、实和复代数几何、凸分析、微分几何、数值分析和最优化理论的技术，以及广泛的数学知识。聂家旺在包括传感器网络和系统控制在内的多个应用领域拥有个人经验，这为他的数学和技术提供了指导。这项建议的其他重要特点是将研究和教育结合起来，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，就如何创造新的研究成果向博士后学者提供建议。

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.