Fast First-Order Methods for Large-Scale Nonconvex Optimization and Semidefinite Programming
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Sujanani, Arnesh M.
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Abstract
This thesis presents fast and efficient first-order methods, with provable complexity bounds, for solving unconstrained convex/strongly convex composite optimization problems, linearly-constrained nonconvex composite optimization problems, and semidefinite programs. We show through extensive computational experiments that the algorithms developed in this thesis are anywhere from 5 to 100 times faster than other existing state-of-the-art codes on important classes of optimization problems such as phase retrieval, matrix completion, sparse PCA, and logistic regression. In the context of semidefinite programming, the algorithm developed in this thesis, namely HALLaR, can also solve (to high accuracy) large-scale problems that have never been solved before due to its low-memory requirements and efficient and highly accurate inner solver. For example, in less than 20 minutes, HALLaR can solve a maximum stable set SDP with 1 million vertices and 10 million edges within 1e-5 relative accuracy while the second-best performing method is unable to find such a solution in 4 hours.
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2024-07-22
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Dissertation