A faster interior-point method for sum-of-squares optimization

Jiang, Shunhua, Natura, Bento and Weinstein, Omri (2022) A faster interior-point method for sum-of-squares optimization. In: Bojanczyk, Mikolaj, Merelli, Emanuela and Woodruff, David P., (eds.) 49th EATCS International Conference on Automata, Languages, and Programming, ICALP 2022. Leibniz International Proceedings in Informatics, LIPIcs. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. ISBN 9783959772358

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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.

Item Type:	Book Section
Official URL:	https://drops.dagstuhl.de/opus/
Additional Information:	© 2022 The Authors
Divisions:	Mathematics
Subjects:	Q Science > QA Mathematics
Date Deposited:	14 Jul 2022 14:39
Last Modified:	11 Dec 2024 18:06
URI:	http://eprints.lse.ac.uk/id/eprint/115561

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