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.
