An introduction to polynomial optimization

Polynomial optimization is a broad class of optimization problems whose objective functions and constraining functions are polynomials. The moment-SOS (sum of squares) hierarchy is the most powerful method to solve polynomial optimization globally. In this paper, we introduce moment-SOS methods for solving polynomial optimization. We also study their finite convergence theory, the certificate for global optimality, and how to extract minimizers. When the standard moment-SOS hierarchy does not have finite convergence, we introduce tight relaxations by using Lagrange multiplier expressions.