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 q²_i be an n-variate SOS polynomial of degree 2d. Denoting by L : = binom(n+d,d) and U : = binom(n+2d,2d) the dimensions of the vector spaces in which q_i’s and p live respectively, our algorithm runs in time Õ(LU^{1.87}). This is polynomially faster than state-of-art SOS and semidefinite programming solvers [Jiang et al., 2020; Huang et al., 2021; Papp and Yildiz, 2019], which achieve runtime Õ(L^{0.5} min{U^{2.37}, L^{4.24}}). 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 [Papp and Yildiz, 2019], 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.
@InProceedings{jiang_et_al:LIPIcs.ICALP.2022.79, author = {Jiang, Shunhua and Natura, Bento and Weinstein, Omri}, title = {{A Faster Interior-Point Method for Sum-Of-Squares Optimization}}, booktitle = {49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)}, pages = {79:1--79:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-235-8}, ISSN = {1868-8969}, year = {2022}, volume = {229}, editor = {Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.79}, URN = {urn:nbn:de:0030-drops-164205}, doi = {10.4230/LIPIcs.ICALP.2022.79}, annote = {Keywords: Interior Point Methods, Sum-of-squares Optimization, Dynamic Matrix Inverse} }
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