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

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 q_i² 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(LU^1.87). This is polynomially faster than state-of-art SOS and semidefinite programming solvers [16, 15, 27], 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 [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.