CAREER: Robust Optimization and Applications

工程中使用的大多数模型都容易出错：对物理参数的测量很差，手头的现象非常复杂，使得“精确”的模型太难处理，等等。这些不确定性现在是优化使用的一个重要限制因素，本质上它倾向于根据问题数据微调解决方案。我提出了一个基于稳健优化方法的具有不确定性的决策问题的一般框架。在这个框架中，问题数据的不确定性被视为确定性的、未知但有界的。稳健的解决方案是允许问题数据在给定范围内发生变化的解决方案。这样的解通常很难计算，但可以设计出基于凸优化的松弛技术来以穷举的方式逼近问题。通过巧妙地利用松弛过程中的不确定性结构，在计算机工作量和精度之间获得了有效的折衷。在反馈控制系统分析和设计领域，稳健性的思想是众所周知的，我和其他人已经发展了基于线性矩阵不等式和相关凸优化的非常有效的松弛方法。显然，稳健优化远远超出了反馈控制的范畴，例如涉及通信网络设计中出现的具有不确定数据的组合优化问题。这开启了新的研究领域，在这里，似乎非常不同的问题被分析和解决在一个统一的框架中。我的建议围绕稳健优化的理论和算法：凸优化松弛，大规模问题，精度/复杂性权衡，软件。这项研究的一个具体目标是开发各种椭球逼近工具，应用于具有结构不确定性的系统的数据拟合和估计、仿真和控制。此外，我试图将这些技术应用于工程中的几个重要设计问题，包括RC电路的时序分析、滤波器和天线阵列设计、通信网络设计等。除了稳健的优化工具和工程应用之外，我计划从稳健性和最坏情况风险的观点研究财务和管理中的几个重要问题，这些概念开始引起社会的极大兴趣。我打算开发快速有效的稳健优化工具，专门针对财务和管理问题，基于椭球技术进行最坏情况的模拟和控制。我的教育计划旨在为我们的学生在优化领域提供坚实的理论和实践基础，强调不确定性和稳健性的关键问题。这一目标将通过两门学科的课程开发来实现：稳健优化高级课程和凸优化基础研究生课程。这项计划的一个重要方面是开发用户友好的稳健优化软件工具，以补充我过去为凸优化开发的工具。

Most models used in engineering are prone to errors: poor measurement of physicalparameters, complexity of phenomena at hand making "exact" models too hard to handle,etc. These uncertainties are now an important limiting factor in the use of optimization,which by nature tends to finely tune solutions to problem data.I propose a general framework for decision problems with uncertainty, based on a robustoptimization approach. In this framework, uncertainty on problem data is treated as deterministic, unknown-but-bounded. A robust solution is one which tolerates changes in theproblem data, up to the given bound. Such a solution is too hard to compute in general, butrelaxation techniques based on convex optimization can be devised to approximate the problem in an exhaustive manner. An efficient trade-off between computer effort and accuracyis obtained by finely exploiting the uncertainty structure in the relaxation process.The idea of robustness is well-known in the area of feedback control systems analysisand design, where I and others have developed very effective relaxation methods based onLinear Matrix Inequalities and related convex optimization. Obviously, robust optimization is relevant far beyond feedback control, for example to the combinatorial optimizationproblems with uncertain data arising in communications network design. This opens newavenues of research where seemingly very different problems are analyzed and solved in aunified framework.My proposal centers around theory and algorithms for robust optimization: convex optmization relaxations, large-scale problems, accuracy/complexity trade-offs, software. Aspecific target of this research is the development ofvarious ellipsoidal approximation tools,with applications to data fitting and estimation, simulation, and control of systems withstructured uncertainty. In addition, I seek to apply these techniques to several importantdesign problems in engineering, including timing analysis of RC circuits, filter and antennaarray design, communications network design, and others.In addition to robust optimization tools and engineering applications, I plan to study sev-eral important problems in finance and management under the viewpoint of robustness andworst-case risk-notions that begin to stir great interest in the community. I intend to develop fast and efficient robust optimization tools that are specific to finance and managementissues, based on ellipsoidal techniques for worst-case simulation and control.My education plans aims at providing a solid theoretical and practical basis for ourstudents in the area of optimization, emphasizing the crucial issues of uncertainty and robustness. This goal will be implemented by course development intwo subjects: an advancedcourse in robust optimization, and a basic graduate course in convex optimization. An important aspect of this plan is the development of user-friendly robust optimization softwaretools that complement those I have developed in the past for convex optimization.