CAREER: Robust Optimization and Applications

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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.