CAREER: Foundations and Algorithms for Nonconvex Nonsmooth Optimization: From Local Solvers to Global Certification

Optimization serves as the mathematical engine powering modern artificial intelligence and complex decision-making systems. Many real-world challenges, however, ranging from managing energy grids to training machine learning models, involve mathematical landscapes that are jagged, unpredictable, and obscured by data noise. These irregularities often trap existing technologies in suboptimal or inefficient solutions. This project pursues a new generation of rigorous mathematical tools and stable algorithms designed to navigate these difficult landscapes with precision and speed. The research will be translated into open-source software to ensure these high-performance tools are accessible to both academic researchers and industry practitioners. Furthermore, educational initiatives will span from K-12 outreach to community college teacher training and specialized graduate instruction, ensuring the next generation of scholars is equipped to tackle the new wave of global scientific and engineering challenges. The research supported by this award is to establish a rigorous theoretical and algorithmic framework for nonconvex and nonsmooth optimization problems, focusing on both the efficient computation of local solutions and the effective certification of global optimality. The research addresses fundamental challenges in problems that lack standard structural assumptions. The technical approach includes: establishing new variational characterizations for irregular objectives to enable principled function approximations; developing stable algorithms capable of escaping irregular saddle points; leveraging implicit dimension reduction induced by nonsmooth maps to create scalable second-order methods; and implementing a novel homotopic sketching framework to provide global optimality certificates via progressively refined relaxations. The project aims to deliver convergence guarantees and numerical algorithms for broad classes of optimization problems that currently lack effective solution methods, thereby advancing the theoretical and algorithmic frontiers of the field. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria. NSF Award ID: 2541022 | Program: 01002627DB NSF RESEARCH & RELATED ACTIVIT,01003031DB NSF RESEARCH & RELATED ACTIVIT,01002930DB NSF RESEARCH & RELATED ACTIVIT | Principal Investigator: Ying Cui | Institution: University of California-Berkeley, BERKELEY, CA | Award Amount: $364,815 View on NSF Award Search: https://www.nsf.gov/awardsearch/show-award/?AWD_ID=2541022 View on Research.gov: https://www.research.gov/awardapi-service/v1/awards/2541022.html