Thumbnail Image
Item

Reducing nonnegativity over general semialgebraic sets to nonnegativity over simple sets

Kuryatnikova,Olga
; 
Vera,Juan C.
; 
Zuluaga,Luis F.
Abstract
A nonnegativity certificate (NNC) is a way to write a polynomial so that its nonnegativity on a semialgebraic set becomes evident. Positivstellensätze (Psätze) guarantee the existence of NNCs. Both NNCs and Psätze underlie powerful algorithmic techniques for optimization. This paper proposes a universal approach to derive new Psätze for general semialgebraic sets from ones developed for simpler sets, such as a box, a simplex, or the nonnegative orthant. We provide several results illustrating the approach. First, by considering Handelman’s Positivstellensatz (Psatz) over a box, we construct non-SOS Schmüdgen-type Psätze over any compact semialgebraic set, that is, a family of Psätze that follow the structure of the fundamental Schmüdgen’s Psatz but where instead of SOS polynomials, any class of polynomials containing the nonnegative constants can be used, such as SONC, DSOS/SDSOS, hyperbolic, or sums of AM/GM polynomials. Second, by considering the simplex as the simple set, we derive a sparse Psatz over general compact sets which does not rely on any structural assumptions of the set. Finally, by considering Pólya’s Psatz over the nonnegative orthant, we derive a new non-SOS Psatz over unbounded sets which satisfy some generic conditions. All these results contribute to the literature regarding the use of non-SOS polynomials and sparse NNCs to derive Psätze over compact and unbounded sets. Throughout the article, we illustrate our results with relevant examples and numerical experiments.
Description
Publisher Copyright: © 2024 Society for Industrial and Applied Mathematics Publications. All rights reserved.
Date
2024-04
Journal Title
Journal ISSN
Volume Title
Publisher
Collections
Tilburg University Research Output Collection
Show all metadata
Files
Thumbnail Image
22m1501027.pdf
Adobe PDF, 1.42 MB
Research Projects
Organizational Units
Journal Issue
Keywords
certificates of nonnegativity, non-SOS polynomials, polynomial optimization, Positivstellensatz, SDSOS polynomials, SONC polynomials, sparsity
Citation
Kuryatnikova, O, Vera, J C & Zuluaga, L F 2024, 'Reducing nonnegativity over general semialgebraic sets to nonnegativity over simple sets', SIAM Journal on Optimization, vol. 34, no. 2, pp. 1970-2006. https://doi.org/10.1137/22M1501027
URI
https://hdl.handle.net/20.500.14602/81799
License
info:eu-repo/semantics/openAccess
cb
Embedded videos