A Faster Interior-Point Method for Sum-Of-Squares Optimization

Shunhua Jiang*, Bento Natura*, Omri Weinstein*

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

3 Scopus citations

Abstract

We present a faster interior-point method for optimizing sum-of-squares (SOS) polynomials, which are a central tool in polynomial optimization and capture convex programming in the Lasserre hierarchy. Let p = ∑ i qi2 be an n-variate SOS polynomial of degree 2d. Denoting by (Equation presented) and (Equation presented) the dimensions of the vector spaces in which qi's and p live respectively, our algorithm runs in time Õ(LU1.87). This is polynomially faster than state-of-art SOS and semidefinite programming solvers [16, 15, 27], which achieve runtime Õ(L0.5 min{U2.37, L4.24}). The centerpiece of our algorithm is a dynamic data structure for maintaining the inverse of the Hessian of the SOS barrier function under the polynomial interpolant basis [27], which efficiently extends to multivariate SOS optimization, and requires maintaining spectral approximations to low-rank perturbations of elementwise (Hadamard) products. This is the main challenge and departure from recent IPM breakthroughs using inverse-maintenance, where low-rank updates to the slack matrix readily imply the same for the Hessian matrix.

Original languageAmerican English
Title of host publication49th EATCS International Conference on Automata, Languages, and Programming, ICALP 2022
EditorsMikolaj Bojanczyk, Emanuela Merelli, David P. Woodruff
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
Pages79:1-79:20
Number of pages20
ISBN (Electronic)9783959772358
DOIs
StatePublished - 1 Jul 2022
Event49th EATCS International Conference on Automata, Languages, and Programming, ICALP 2022 - Paris, France
Duration: 4 Jul 20228 Jul 2022

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume229
ISSN (Print)1868-8969

Conference

Conference49th EATCS International Conference on Automata, Languages, and Programming, ICALP 2022
Country/TerritoryFrance
CityParis
Period4/07/228/07/22

Bibliographical note

Publisher Copyright:
© Shunhua Jiang, Bento Natura, and Omri Weinstein; licensed under Creative Commons License CC-BY 4.0

Keywords

  • Dynamic Matrix Inverse
  • Interior Point Methods
  • Sum-of-squares Optimization

Fingerprint

Dive into the research topics of 'A Faster Interior-Point Method for Sum-Of-Squares Optimization'. Together they form a unique fingerprint.

Cite this