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

Authors Shunhua Jiang, Bento Natura, Omri Weinstein

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Shunhua Jiang
  • Columbia University, New York, NY, USA
Bento Natura
  • London School of Economics, UK
Omri Weinstein
  • The Hebrew University, Jerusalem, Israel
  • Columbia University, New York, NY, USA


The second author would like to thank Vissarion Fisikopoulos and Elias Tsigaridas for introducing him from a practical perspective to Sum-of-Squares Optimization under the interpolant basis.

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Shunhua Jiang, Bento Natura, and Omri Weinstein. A Faster Interior-Point Method for Sum-Of-Squares Optimization. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 79:1-79:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)


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.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Continuous functions
  • Mathematics of computing → Convex optimization
  • Mathematics of computing → Semidefinite programming
  • Mathematics of computing → Stochastic control and optimization
  • Interior Point Methods
  • Sum-of-squares Optimization
  • Dynamic Matrix Inverse


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