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Explore the Moment-SOS (Sum of Squares) hierarchy and its wide-ranging applications in this comprehensive lecture delivered at the International Conference on Boolean Satisfiability 2025. Delve into the mathematical foundations of the moment-SOS hierarchy, a powerful computational framework that bridges semidefinite programming with polynomial optimization problems. Learn how this hierarchy provides systematic approximation schemes for solving challenging optimization problems involving polynomial constraints and objectives. Discover the theoretical underpinnings of moment relaxations and their connection to sum-of-squares decompositions, understanding how these tools enable the transformation of non-convex polynomial optimization into tractable semidefinite programs. Examine practical applications across diverse fields including control theory, robotics, machine learning, and combinatorial optimization, where the moment-SOS hierarchy has proven instrumental in solving previously intractable problems. Gain insights into the computational aspects of implementing these methods and understand the trade-offs between approximation quality and computational complexity as you progress through higher levels of the hierarchy.
