Variational Analysis for Practical Optimization

刘易斯DMS-0806057研究人员和他的学生研究变分分析、直观的、可能是随机的、通用的非光滑优化算法与具体模型应用之间的相互作用，特别是在鲁棒控制中。一种范式，起源于80年代S的德梅尔，与条件作用、病态和计算速度有关，启发了研究者的思考。对矩阵伪谱的研究有力地说明了一般方法：暂态动力学行为的基本指标，伪谱具有丰富的结构、混合复形和矩阵分析、半代数几何和数值线性代数，这些都是伪谱优化的组成部分。尤其是半代数几何正在发展成为一种基本工具，不仅在伪谱等应用中，而且在变分分析中也是如此。例如，研究者使用这种方法来研究新的结构工具，如“部分光滑性”，从而分析在计算实践中观察到的收敛速度。研究人员方法的基础是变分分析、经典数学、数值计算和应用建模的复杂结合。通常，复杂工程系统的设计者会改变某些功能，以优化系统性能的某些方面。减震器就是一个简单的例子：通过改变它所含润滑剂的等级，制造商可以改变悬挂系统的感觉有多硬。在这样一个物理、电子、通信或信息系统中找到抑制振动的最佳选择是“不顺利的”：选择对系统的微小变化非常敏感，这是它们似乎很难计算的原因之一。研究人员通过将数学建模和分析与科学计算相结合，促进了这一重要但具有挑战性的优化过程。他通过说明性和技术性出版物、备受瞩目的讲座和在线向不同的国际科学和工程受众大力传播他的研究。作为该项目不可或缺的一部分，研究人员的博士生开发关键技能，在复杂的数学理论和许多非光滑优化的实际应用之间架起一座桥梁。

LewisDMS-0806057 The investigator and his students study the interplaybetween variational analysis, intuitive, possibly randomized,general-purpose nonsmooth optimization algorithms, andapplications to concrete models, particularly in robust control. A paradigm, originating with Demmel in the 1980's, relatingconditioning, ill-posedness, and computational speed, inspiresthe investigator's thinking. The study of matrix pseudospectrapowerfully illustrates the general technique: fundamentalindicators of transient dynamical behavior, pseudospectra have arich structure, blending complex and matrix analysis,semi-algebraic geometry, and numerical linear algebra, allingredients in pseudospectral optimization. Semi-algebraicgeometry in particular is developing into a fundamental tool notjust in applications such as pseudospectra, but throughoutvariational analysis. The investigator uses this approach, forexample, to study novel structural tools such as "partialsmoothness", thereby analyzing convergence rates observed incomputational practice. Fundamental to the investigator'sapproach is a complex blend of variational analysis, classicalmathematics, numerical computation, and applied modeling. Often the designer of a complex engineering system variescertain features in order to optimize some aspect of systemperformance. A simple example is a shock-absorber: by varyingthe grade of lubricant it contains, the manufacturer can modifyhow stiff a suspension system feels. Finding optimal choices fordamping vibrations in such a physical, electronic, communicationsor information system is "nonsmooth": choices are very sensitiveto slight changes in the system, one reason they seem hard tocompute. The investigator facilitates this important butchallenging optimization process by blending mathematicalmodeling and analysis with scientific computation. Hedisseminates his research energetically through expository andtechnical publications, high-profile lectures, and online, todiverse international scientific and engineering audiences. Asan integral part of the project, the investigator's PhD studentsdevelop crucial skills bridging divisions between sophisticatedmathematical theory and the many practical applications ofnonsmooth optimization.