Collaborative Research: AF: Medium: Polynomial Optimization: Algorithms, Certificates and Applications

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Computational problems arising in diverse fields of sciences and engineering can be modeled as optimizing an appropriate objective function subject to a set of constraints. A case of wide interest that captures a surprising array of problems is when the objective function is a polynomial of low-degree. A rich body of theoretical and applied work has led to a fairly extensive understanding of algorithms and hardness for optimizing linear and quadratic functions on domains such as the unit sphere or the hypercube in high dimensions. The situation for polynomials of degree greater than two is, however, not yet well understood. The goal of this project is to advance the frontiers of optimizing higher-degree polynomials in terms of algorithms to estimate and proofs to approximately bound their optima, and then leverage this enhanced understanding in diverse applications. The motivation is both the intrinsic importance of polynomial optimization, as well as several extraneous contexts (constraint satisfaction, graph theory, high-dimensional geometry, proof complexity, and pseudo-randomness, to name a few) where polynomial/tensor optimization arises naturally and could hold the key to further progress. An an example direction, of high importance in modern learning and inference applications, is the generalization of the frequently used principal-component analysis of matrix-valued data to higher-order tensors.This project presents three carefully crafted and intertwined directions to significantly advance the understanding of polynomial optimization. This includes a fresh approach to finding new rounding algorithms that will lead to approximation algorithms with improved guarantees for maximizing cubic and higher-degree polynomials, which in turn is expected to lead to progress beyond longstanding barriers for discrete problems such as Maximum Cut or Small Set Expansion on graphs. The project also involves new approaches towards hardness results for approximate polynomial optimization; currently only very weak bounds are known, and there is a huge gap between the known algorithmic and hardness results. Third, with impetus provided by some recent work by the investigators on refuting constraint-satisfaction problems, the project will embark on a study of polynomial optimization through the lens of certificates on their optima, extending beyond the state of the art linear-algebraic and spectral certificates. Such certificates could have significant ramifications in pseudo-randomness, producing "certified random objects" that are functionally as good as the gold standard (but often highly elusive) explicit constructions. The research and outreach activities of the project will build bridges to allied research communities in algebraic geometry, statistics, operations research, signal processing, and machine learning. The project investigators will train and mentor several graduate students, and also provide engaging research experiences to undergraduates. The research findings will inform graduate level courses on approximate optimization by unifying several problems under the umbrella of polynomial optimization.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.