Geometry in nonsmooth optimization

非光滑优化中的几何

• 批准号: 1208338
• 负责人: Adrian Lewis
• 金额: $ 41.3万
• 依托单位: Cornell University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-09-01 至 2016-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1208338&HistoricalAwards=false
• 关键词: Geometry; nonsmooth; optimization

While nonsmooth optimization is ubiquitous across science and engineering, variational analysis - its elegant mathematical foundation - has achieved narrower practical impact than it warrants. This project attacks that deficit through "structure": not in the traditional explicit computational sense, but rather in the sense of intrinsic geometry. The investigator studies two overlapping mathematical strategies. The first uses semi-algebraic geometry as a rich and natural model for the world of concrete optimization problems. In that world, much of the technicality and pathology obscuring variational analysis for practitioners is transformed, leaving powerful stratification tools and simple access to "generic" properties in optimization. The second strategy emphasizes partial smoothness, a powerful geometric property originating from the theory of optimality conditions and sensitivity analysis, but also in perfect resonance with active-set algorithms. With these theoretical tools at hand, the investigator focuses foremost on two fresh and promising computational methods for nonsmooth optimization. The first is, counter-intuitively, just the classical BFGS method for smooth optimization, mildly adjusted for the nonsmooth world. BFGS is simple, intuitive, general-purpose, much easier to implement successfully than traditional "bundle" methods, and broadly effective in practical applications, notably in robust control. Mysteriously, BFGS always (essentially) seems to converge linearly and to identify partly smooth structure. The investigator seeks explanations. The second focal computational method is a proximal algorithm for composite optimization that is simple, versatile, and, in contrast with BFGS, well-grounded theoretically. This algorithm has proved successful on huge problems, such as compressed sensing, but is potentially slow. Partial smoothness will strengthen convergence in theory, and speed it in practice.The broader significance and importance of this project derive from the investigator's transformative goal of bridging the gulf between mathematical theory and data-driven practice in resource allocation problems beyond the reach of traditional calculus. A particularly important example is the kind of robust control engineering underlying applications like modern aircraft electronics. The investigator builds on a strong track-record of high-calibre publications, innovative scholarship and outreach, and intellectual leadership. PhD students based in Cornell's highly ranked School of ORIE (where the investigator is Director) will engage all aspects of the project, publishing and presenting at professional meetings; the investigator will engage more broadly through seminars and his award-winning teaching, as well as through graduate texts, survey articles, multidisciplinary collaboration, and prominent international lectures to broad scientific and engineering audiences.

虽然非光滑优化在科学和工程中无处不在，但变分分析-其优雅的数学基础-所取得的实际影响比它所保证的要小。 这个项目通过“结构”来解决这个问题：不是在传统的显式计算意义上，而是在内在几何意义上。 研究人员研究了两个重叠的数学策略。 第一个使用半代数几何作为一个丰富和自然的模型，为世界的具体优化问题。 在这个世界中，许多技术性和病理学模糊变分分析的从业者被改造，留下强大的分层工具和简单的访问“通用”的优化性能。 第二种策略强调部分光滑，一个强大的几何属性起源于理论的最优性条件和灵敏度分析，但也在完美的共振与活动集算法。 有了这些理论工具在手，调查人员首先集中在两个新的和有前途的计算方法的非光滑优化。 第一个是，与直觉相反，只是用于平滑优化的经典BFGS方法，针对非平滑世界进行了适度调整。 BFGS是简单，直观，通用，更容易成功地实现比传统的“捆绑”方法，并广泛有效的实际应用中，特别是在鲁棒控制。 不幸的是，BFGS似乎总是（基本上）线性收敛，并识别部分光滑结构。 调查员寻求解释。 第二个焦点计算方法是一个近似算法的复合材料优化，是简单的，通用的，并与BFGS相比，良好的理论基础。 这种算法已经被证明是成功的巨大的问题，如压缩感知，但可能是缓慢的。 部分平滑将加强理论上的收敛，并在实践中加速收敛。该项目的更广泛的意义和重要性来自研究者的变革目标，即弥合数学理论与数据驱动的资源分配问题之间的鸿沟，超越传统微积分的范围。 一个特别重要的例子是现代飞机电子设备等应用的鲁棒控制工程。 该研究员建立在高质量出版物，创新奖学金和外联以及知识领导力的良好记录之上。 康奈尔大学排名靠前的ORIE学院的博士生（研究员是主任）将参与项目的各个方面，在专业会议上发表和演讲;研究员将通过研讨会和他获奖的教学，以及通过研究生文本，调查文章，多学科合作和着名的国际讲座更广泛地参与广泛的科学和工程观众。