  We present a novel, general, and unifying point of view on sparse approaches
to polynomial optimization. Solving polynomial optimization problems to global
optimality is a ubiquitous challenge in many areas of science and engineering.
Historically, 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 to understand existing approaches from different schools of
thought, to develop novel relaxation schemes, and to derive a flexible duality
theory, which can be specialized to many concrete situations that have been
considered in the literature. We unify different approaches to polynomial
optimization including polyhedral approximations, dense semidefinite
relaxations, SONC, SAGE, and TSSOS in a self-contained exposition. We also
carry out computational experiments to demonstrate the practical advantages of
a flexible usage of pattern-based sparse relaxations of polynomial optimization
problems.
