Geometry in nonsmooth optimization

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”来弥补这一缺陷：不是在传统的明确计算意义上，而是在内在几何意义上。研究者研究了两种重叠的数学策略。第一种是使用半代数几何作为一个丰富而自然的模型来解决具体的优化问题。在那个世界里，许多技术和病理学模糊了从业者的变分分析，留下了强大的分层工具和简单的访问优化中的“通用”属性。第二种策略强调部分平滑，这是一种源自最优性条件理论和灵敏度分析的强大几何特性，但也与主动集算法完美共振。有了这些理论工具，研究者首先关注两种新的和有前途的非光滑优化计算方法。与直觉相反，第一种方法只是经典的BFGS平滑优化方法，对非光滑世界进行了轻微调整。BFGS简单、直观、通用，比传统的“捆绑”方法更容易成功实现，并且在实际应用中广泛有效，特别是在鲁棒控制中。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。调查者寻求解释。第二种焦点计算方法是一种用于复合优化的近似算法，它简单、通用，与BFGS相比，具有良好的理论基础。该算法已被证明在诸如压缩感知等重大问题上是成功的，但可能速度较慢。部分平滑在理论上加强了收敛性，在实践中加快了收敛性。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中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等应用程序的鲁棒控制工程。研究者建立在高水平的出版物，创新的奖学金和推广，以及智力领导的良好记录。康奈尔大学排名靠前的ORIE学院的博士生（研究者是该学院的主任）将参与项目的各个方面，发表论文并在专业会议上发表；研究者将通过研讨会和他的获奖教学，以及研究生文本，调查文章，多学科合作，以及面向广泛的科学和工程受众的著名国际讲座，更广泛地参与。

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.