Monomial patterns in polynomial optimization

Abstract

Convexification is a core technique in global polynomial optimization, which is used to generate convex relaxations of a polynomial optimization problem (POP). These relaxations in turn allow to compute bounds on the optimal value of a POP. Currently, there are two main convexification approaches competing in theory and practice: the approach of nonlinear programming and the approach based on positivity certificates from real algebra. The former is comparatively cheap from a computational point of view, but typically does not provide tight relaxations with respect to bounds for the original problem. The latter is typically computationally expensive, but provides tight relaxations. We embed both kinds of approaches into a unified framework of monomial relaxations. This framework of monomial relaxations is based on groups of exponents, which we call patterns. In order to build a relaxation, the POP is linearized by replacing each of its monomials with a monomial variable. Then the monomial variables that are indexed by the exponents of a pattern are linked by convex constraints. By identifying the appropriate patterns and their associated constraints, a variety of established convexification methods can be expressed within this framework. These include convexification methods based on sum-of-squares polynomials or nonnegative circuit polynomials as as well as multilinear envelopes. Within our framework we can freely combine the different patterns and their constraints. The combination of the different patterns allows to exploit the monomial structure of the polynomial problem. By selecting appropriate combinations of patterns we can trade off the quality of the bounds against computational expenses. Thus, it is possible to develop custom-made convexification strategies that are fitted to the problem structure and the demands of the user. Examples of different such strategies are given. Furthermore, we develop a new pattern type called truncated submonoid and determine the corresponding convex constraints. Different relaxations that are derived from combinations of patterns are numerically tested on self-generated benchmark instances. The computational experiments yield very encouraging results.