Monomial-wise optimal separable underestimators for mixed-integer polynomial optimization

Christoph Buchheim () and Claudia D’Ambrosio
Additional contact information
Christoph Buchheim: TU Dortmund
Claudia D’Ambrosio: École Polytechnique

Journal of Global Optimization, 2017, vol. 67, issue 4, No 3, 759-786

Abstract: Abstract We introduce a new method for solving box-constrained mixed-integer polynomial problems to global optimality. The approach, a specialized branch-and-bound algorithm, is based on the computation of lower bounds provided by the minimization of separable underestimators of the polynomial objective function. The underestimators are the novelty of the approach because the standard approaches in global optimization are based on convex relaxations. Thanks to the fact that only simple bound constraints are present, minimizing the separable underestimator can be done very quickly. The underestimators are computed monomial-wise after the original polynomial has been shifted. We show that such valid underestimators exist and their degree can be bounded when the degree of the polynomial objective function is bounded, too. For the quartic case, all optimal monomial underestimators are determined analytically. We implemented and tested the branch-and-bound algorithm where these underestimators are hardcoded. The comparison against standard global optimization and polynomial optimization solvers clearly shows that our method outperforms the others, the only exception being the binary case where, though, it is still competitive. Moreover, our branch-and-bound approach suffers less in case of dense polynomial objective function, i.e., in case of polynomials having a large number of monomials. This paper is an extended and revised version of the preliminary paper [4].

Keywords: Polynomial optimization; Mixed-integer nonlinear programming; Separable underestimators (search for similar items in EconPapers)
Date: 2017
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (1)

Downloads: (external link)
http://link.springer.com/10.1007/s10898-016-0443-3 Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:jglopt:v:67:y:2017:i:4:d:10.1007_s10898-016-0443-3

Ordering information: This journal article can be ordered from
http://www.springer. ... search/journal/10898

DOI: 10.1007/s10898-016-0443-3

Access Statistics for this article

Journal of Global Optimization is currently edited by Sergiy Butenko

More articles in Journal of Global Optimization from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().