Solving SDP Faster: A Robust IPM Framework and Efficient Implementation

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

“…and can thus be solved using off-the-shelf SDP solvers. However, despite recent breakthroughs on the runtime of general SDP solvers via interior-point methods (IPMs) [16,15], this SDP reformulation of (3) does not scale well for moderately large degrees, i.e., whenever U ≪ L 2 in (4). This is because the SDP reformulation always incurs a factor of at least L 2 , even when U ≪ L 2 , as this is the SDP variable size (the PSD matrix X has size L × L).…”

“…This is because the SDP reformulation always incurs a factor of at least L 2 , even when U ≪ L 2 , as this is the SDP variable size (the PSD matrix X has size L × L). Indeed, for the current fast-matrix-multiplication (FMM) exponent ω ≈ 2.37 [21,1], the running time of state-of-the-art SDP solvers [16,15] for SOS optimization (Problem 3) is 2…”

“…The main challenge here, compared to previous LP and SDP solvers [37,22,9,16,15], is that low-rank updates to M do not readily translate to a low-rank update to (M •2 ) −1 , since Hadamard-products can increase the rank rk(A • B) ≤ rk(A) • rk(B), in contrast to standard matrix multiplication which does not increase the rank rk(AB) ≤ max{rk(A), rk(B)}. This means that we cannot directly apply Woodbury's identity to efficiently update the inverse of the Hessian, which is the common approach in all aforementioned works.…”

“…Denoting by L := n+d d and U := n+2d 2d 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 lowrank 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.…”

“…Denoting by L := n+d d and U := n+2d 2d 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 }).…”