# Error bounds for monomial convexification in polynomial optimization

## TL;DR

This paper provides explicit error bounds for convexifying monomials in polynomial optimization, enabling better understanding of approximation quality in convex relaxations.

## Contribution

It introduces the first comprehensive error bounds for monomial convexification over subsets of [0,1]^n, including multilinear cases and convex hull derivations.

## Key findings

- Error bounds depend mainly on monomial degree
- Bounds are computationally straightforward to evaluate
- Convex hull of multilinear monomials over [-1,1]^n derived

## Abstract

Convex hulls of monomials have been widely studied in the literature, and monomial convexifications are implemented in global optimization software for relaxing polynomials. However, there has been no study of the error in the global optimum from such approaches. We give bounds on the worst-case error for convexifying a monomial over subsets of $[0,1]^n$. This implies additive error bounds for relaxing a polynomial optimization problem by convexifying each monomial separately. Our main error bounds depend primarily on the degree of the monomial, making them easy to compute. Since monomial convexification studies depend on the bounds on the associated variables, in the second part, we conduct an error analysis for a multilinear monomial over two different types of box constraints. As part of this analysis, we also derive the convex hull of a multilinear monomial over $[-1,1]^n$.