Optimization Methods for Nonconvex Structured Optimization

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This project will advance fundamental algorithmic theory and software tools for solving optimization problems with wide applications in science, engineering and industry. Specifically, the project will be in the area of structured nonconvex nonlinear optimization, a critical component in many modern applications ranging from signal/image processing, real-time optimal control to stochastic learning. The project aims to develop algorithms with focus on the following features: speed, problem dependence, and ease of use for researchers in both optimization and computational data science community. Students will be involved and will have opportunities for interdisciplinary research. Software will be developed.This project will develop theoretically strong and numerically efficient algorithms as well as the software for solving nonconvex structured optimization. These algorithms will solve the subproblems inexactly with guaranteed global convergence as well as feature an optimal computational complexity when the problem features convexity structure. The algorithms will be based on recent work on proximal and stochastic gradient methods for structured composite minimization, inexact alternating direction multiplier methods (ADMM) for separable convex/nonconvex optimization and active set methods for polyhedral constrained optimization. In addition, second-order techniques for accelerating the convergence will be also explored.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.