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普遍存在，但变分分析（其优雅的数学基础）所取得的实际影响比其应有的要小。 该项目通过“结构”来解决这一缺陷：不是传统的显式计算意义上的，而是内在几何意义上的。 研究者研究两种重叠的数学策略。 第一个使用半代数几何作为具体优化问题领域的丰富而自然的模型。 在那个世界中，许多使从业者变得模糊的技术细节和病态变分分析发生了变化，留下了强大的分层工具和对优化中“通用”属性的简单访问。 第二种策略强调部分平滑，这是一种强大的几何特性，源自最优条件和灵敏度分析理论，但也与活动集算法完美共鸣。 有了这些理论工具，研究人员首先关注两种新颖且有前途的非光滑优化计算方法。 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轻微调整。 BFGS 简单、直观、通用，比传统的“捆绑”方法更容易成功实施，并且在实际应用中广泛有效，特别是在鲁棒控制方面。 神秘的是，BFGS 似乎总是（本质上）线性收敛并识别部分平滑的结构。 调查员寻求解释。 第二种焦点计算方法是一种复合优化的近端算法，它简单、通用，并且与 BFGS 相比，具有良好的理论基础。 事实证明，该算法在压缩感知等重大问题上是成功的，但速度可能很慢。 部分平滑将加强理论上的收敛，并在实践中加速收敛。该项目更广泛的意义和重要性源于研究者的变革目标，即在传统微积分无法解决的资源分配问题上弥合数学理论和数据驱动实践之间的鸿沟。 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。 该研究人员建立在高水平出版物、创新学术和推广以及知识领导力方面的良好记录之上。 康奈尔大学排名靠前的 ORIE 学院（研究者担任主任）的博士生将参与该项目的各个方面，包括在专业会议上发表和演示； 研究者将通过研讨会和获奖教学、研究生论文、调查文章、多学科合作以及向广大科学和工程受众举办的著名国际讲座来更广泛地参与。