Variational Analysis for Practical Optimization

实际优化的变分分析

• 批准号: 0806057
• 负责人: Adrian Lewis
• 金额: $ 38.79万
• 依托单位: Cornell University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-09-01 至 2012-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0806057&HistoricalAwards=false
• 关键词: Variational; Analysis; Practical; Optimization

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.

LewisDMS-0806057 调查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之间的相互作用，并应用到具体的模型，特别是在鲁棒控制。一个范例，起源于Demmel在20世纪80年代，relatingconditioning，不适定性，和计算速度，激发调查员的思考。 矩阵伪谱的研究有力地说明了一般技术：瞬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，数值线性代数，伪谱优化的所有成分。 特别是半代数几何学正在发展成为一个基本的工具，不仅在伪谱等应用中，而且在变分分析中。 研究人员使用这种方法，例如，研究新的结构工具，如“partialsmoothness”，从而分析收敛率在计算实践中观察到的。 基本调查的方法是一个复杂的混合变分分析，classicalmathematics，数值计算和应用建模。 通常，复杂工程系统的设计者改变某些特性，以优化系统性能的某些方面。 一个简单的例子是减震器：通过改变它所包含的润滑剂的等级，制造商可以改变悬挂系统的刚性。 在这样一个物理、电子、通信或信息系统中寻找最佳选择来阻尼振动是“非光滑的”：选择对系统中的微小变化非常敏感，这是它们似乎难以计算的原因之一。 研究人员通过将数学建模和分析与科学计算相结合来促进这一重要但具有挑战性的优化过程。 他通过临时和技术出版物、高知名度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成果。 作为该项目的一个组成部分，研究者的博士生开发的关键技能之间的桥梁划分抽象的数学理论和非光滑优化的许多实际应用。