Unifying view on sparse convex relaxations in polynomial optimization

We present a novel, general, and unifying point of view and use it to surveys parse and dual approaches to polynomial optimization. Solving polynomial optimization problems to global optimality is a ubiquitous challenge in many areas of science and engineering. Different approaches on how to solve nonconvex polynomial optimization problems based on convex relaxations have been developed in different scientific communities. Here, we introduce the concept of monomial patterns. A pattern determines what monomials are to be linked by convex constraints in a convex relaxation of a polynomial optimization problem. This concept helps understanding existing approaches from different schools of thought, developing novel relaxation schemes, and deriving a flexible duality theory, which can be specialized to many concrete situations that have been considered in the literature. We survey different approaches to polynomial optimization including polyhedral approximations, dense semidefinite relaxations, SONC, SAGE, and TSSOS in a self-contained, unifying exposition.