Error bounds for monomial convexification in polynomial optimization

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 .

The convex hull of a quadratic constraint over a polytope

“…Let S and P be defined as in (1). Next, we show that we may assume without loss of generality that P is full dimension.…”

“…In this case, a constraint of the form f (x) = b is replaced with f (x) ≤ b and f (x) ≥ b, where f is a convex lower approximation and f is a concave upper approximation of f . While there have been a lot of work in function convexification (see for instance [3,49,5,46,35,10,38,6,8,7,41,18,47,45,39,55,56,36,12,16,1,27,51]) it is well-known that it does not necessarily yield the convex hull of the set {x | f (x) = b}. To the best of our knowledge, there have been much less work on explicit convexification of sets: [54,42,43,53,25,32,44,17,34,13].…”

A study of rank-one sets with linear side constraints and application to the pooling problem

“…An important area of study regarding convexification schemes for QCQPs is to convexify commonly occurring substructures, like in the case of integer programming. However, most of the work in this direction in the global optimization area has been focused on convexification of functions (i.e., finding convex and concave envelopes), see for example [4,46,33,10,36,9,8,38,21,48,45,37,52,53,16,19,1,26]. There are relatively lesser number of results on convexification of sets [51,41,42,50,24,30,44,20,32,17,40,23].…”