Nonsmooth Optimization: Structure, Geometry, and Conditioning

本研究项目涉及优化理论，这是一门具有广泛适用性的经典数学学科，随着当前计算发展和挑战的需求而不断发展。应用科学和工程领域的从业者现在更像登山者，而不是山间步行者，他们在陡峭而非平坦的战略景观中探索优化目标。在控制工程、当代统计学或大数据应用中解决这些问题已经产生了变革性的影响。然而，这种成功背后的基本几何原理常常被计算迷雾所掩盖：数学专家通常计算得很少，相反，从事重要科学和工程应用的从业者通常不了解基本原理。该项目旨在弥合这一鸿沟，开发一种几何和计算的创新组合。博士生将参与研究的各个方面。该项目设想了一个统一的数学策略，基于两个对偶但等效的观点：部分平滑的几何思想和识别的算法思想。利用现代变分分析的力量，该项目旨在阐明部分光滑几何如何鼓励具有理想结构（如稀疏性或低秩）的解决方案，流行的当代算法如何因此被吸引（或“识别”）这样的解决方案，因此方法收敛的速度有多快，以及我们如何加速它们。在这个项目的聚光灯下是两个激励算法，它们在计算实践中都很有前途。第一种方法是准线性方法，它可以解决大规模的结构问题，这些问题的显式部分光滑几何结构可能被它所利用。第二种是光滑的准牛顿方法，在非光滑优化方面取得了鲁棒但令人困惑的成功，它不受任何显式几何的影响，但受显式几何的强烈影响。该项目成功的关键将是与经典数学其他领域的相互作用；矩阵分析具有丰富的潜在应用——该项目特别针对“Crouzeix猜想”。从基础的角度来看，常见的多项式不等式可以通过分层形成光滑的表面来诱导部分光滑，使该项目沉浸在半代数几何的基础中。

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.