# Convexification of box-constrained polynomial optimization problems via monomial patterns

## TL;DR

This paper introduces a unified framework for convexifying box-constrained polynomial optimization problems using monomial relaxations, balancing computational cost and relaxation tightness, with promising experimental results.

## Contribution

It develops a novel convexification strategy that unifies nonlinear programming and positivity certificate approaches within a monomial relaxation framework.

## Key findings

- The method effectively balances relaxation quality and computational effort.
- Computational experiments demonstrate promising results.
- The framework offers a flexible trade-off between bound tightness and computational cost.

## Abstract

Convexification is a core technique in global polynomial optimization. Currently, there are two main approaches competing in theory and practice: the approach of nonlinear programming and the approach based on positivity certificates from real algebra. The former are comparatively cheap from a computational point of view, but typically do not provide tight relaxations with respect to bounds for the original problem. The latter are typically computationally expensive, but do provide tight relaxations. We embed both kinds of approaches into a unified framework of monomial relaxations. We develop a convexification strategy that allows to trade off the quality of the bounds against computational expenses. Computational experiments show very encouraging results.