CAREER: Robust Optimization and Applications

大多数模型在工程中使用的是容易出现错误：poor measurement of physicalparameters，复杂的现象在手，使“精确”的模型太难处理，等等，这些不确定性现在是一个重要的限制因素，在使用优化，这是自然倾向于微调解决方案的问题data.I提出了一个总体框架，决策问题的不确定性，基于一个robustoptimization方法。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。一个鲁棒的解决方案是一个容忍的问题数据的变化，直到给定的界限。这样的解决方案是太难计算一般，但松弛技术的基础上凸优化可以设计近似的问题，在一个详尽的方式。通过在松弛过程中精细地利用不确定性结构，获得了计算机工作量和精度之间的有效折衷。鲁棒性的思想在反馈控制系统分析和设计领域中是众所周知的，其中我和其他人已经开发了非常有效的基于线性矩阵不等式和相关凸优化的松弛方法。显然，鲁棒优化的相关性远远超出反馈控制，例如，在通信网络设计中出现的不确定数据的组合优化问题。这开启了新的研究领域，在统一的框架中分析和解决看似非常不同的问题。我的建议围绕鲁棒优化的理论和算法：凸优化松弛，大规模问题，准确性/复杂性权衡，软件。本研究的一个具体目标是开发各种椭球近似工具，应用于结构不确定性系统的数据拟合和估计、仿真和控制。此外，我还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。除了强大的优化工具和工程应用外，我计划从稳健性和最坏情况风险的角度来研究财务和管理中的几个重要问题-这些观念开始在社区中激起极大的兴趣。我打算开发快速和有效的鲁棒优化工具，具体到金融和管理系统，基于椭球技术的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，强调不确定性和鲁棒性的关键问题。这一目标将通过两个学科的课程开发来实现：鲁棒优化高级课程和凸优化基础研究生课程。该计划的一个重要方面是开发用户友好的强大的优化软件工具，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.