Nonsmooth Optimization: Structure, Geometry, and Conditioning

非光滑优化：结构、几何形状和条件

• 批准号: 1613996
• 负责人: Adrian Lewis
• 金额: $ 34.94万
• 依托单位: Cornell University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-09-01 至 2020-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1613996&HistoricalAwards=false
• 关键词: Nonsmooth; Optimization; Structure; Geometry; Conditioning; Nonsmooth; Optimization; Structure; Geometry; Conditioning

This research project concerns optimization theory, a classical mathematical subject of wide applicability that is evolving in response to the demands of current computational developments and challenges. Practitioners across the applied sciences and engineering often now resemble mountaineers more than hill-walkers, exploring optimization goals in sharp rather than smooth strategic landscapes. Solving such problems in control engineering, contemporary statistics, or big data applications has had transformative impact. Often lost in the computational fog, however, has been the fundamental geometry underlying this success: mathematical specialists often compute little, and conversely, practitioners across vital science and engineering applications are typically unaware of the fundamentals. This project aims to bridge that divide, developing an innovative mix of geometry and computation. Ph.D. students will be involved in all aspects of the research.The project envisages a unifying mathematical strategy based on two dual but equivalent viewpoints: the geometric idea of partial smoothness and the algorithmic idea of identification. Using the power of modern variational analysis, the project aims to illuminate how partly smooth geometry encourages solutions with desirable structure (like sparsity or low rank), how popular contemporary algorithms are hence drawn to (or "identify") such solutions, how fast the methods therefore converge, and how we might accelerate them. In the project's spotlight are two motivating algorithms, both very promising in computational practice. The first, a prox-linear method, solves large-scale structured problems whose explicit partly smooth geometry it could potentially exploit. The second, a smooth quasi-Newton method with robust but baffling success for nonsmooth optimization, is blind to any explicit geometry, but is strongly influenced by it. Crucial to the project's success will be interplay with other areas of classical mathematics; matrix analysis is rich in potential applications -- the project aims in particular at the "Crouzeix conjecture." From a foundational perspective, commonly-occurring polynomial inequalities can induce partial smoothness through stratification into smooth surfaces, immersing this project in the fundamentals of semi-algebraic geometry.

该研究项目涉及优化理论，这是一个广泛适用性的经典数学主题，该主题是根据当前计算发展和挑战的需求而发展的。现在，应用科学和工程的从业人员现在通常比山地走者更像登山者，以敏锐而不是平稳的战略景观探索优化目标。在控制工程，当代统计或大数据应用中解决此类问题具有变革性的影响。然而，在计算雾中通常会丢失，这是这一成功的基本几何形状：数学专家通常很少计算，相反，重要的科学和工程应用程序中的从业者通常不知道基本面。该项目旨在弥合分歧，开发几何和计算的创新组合。博士该项目将参与研究的各个方面。该项目设想了一个基于两个双重但等效的观点的统一数学策略：部分平滑度的几何思想和识别算法观念。该项目利用现代变分分析的力量来阐明部分平滑的几何形状如何鼓励具有理想结构（例如稀疏或低级）的解决方案，因此，当代算法如何被吸引到（或识别）此类解决方案，方法的速度以及我们如何加速它们的方法。在项目的聚光灯下，有两个激励算法，在计算实践中都非常有前途。首先是一种代理线性方法，解决了大规模结构化问题，其明确的部分平滑的几何形状可能会利用。第二个，一种平滑的准牛顿方法，具有强大但令人困惑的非平滑优化成功成功，对任何明确的几何形状视而不见，但受到它的影响很大。对该项目的成功至关重要的是与其他古典数学领域的相互作用；矩阵分析具有丰富的潜在应用 - 该项目尤其是针对“ Crouzeix猜想”。从基础的角度来看，通常存在的多项式不等式可以通过分层为平滑表面诱导部分平滑度，从而将该项目浸入半代数几何的基本原理中。