Positive Semidefinite Programming: Mixed, Parallel, and WidthIndependent
Abstract
We give the first approximation algorithm for mixed packing and covering semidefinite programs (SDPs) with polylogarithmic dependence on width. Mixed packing and covering SDPs constitute a fundamental algorithmic primitive with recent applications in combinatorial optimization, robust learning, and quantum complexity. The current approximate solvers for positive semidefinite programming can handle only pure packing instances, and technical hurdles prevent their generalization to a wider class of positive instances. For a given multiplicative accuracy of $\epsilon$, our algorithm takes $O(\log^3(nd\rho) \cdot \epsilon^{3})$ parallelizable iterations, where $n$, $d$ are dimensions of the problem and $\rho$ is a width parameter of the instance, generalizing or improving all previous parallel algorithms in the positive linear and semidefinite programming literature. When specialized to pure packing SDPs, our algorithm's iteration complexity is $O(\log^2 (nd) \cdot \epsilon^{2})$, a slight improvement and derandomization of the stateoftheart (AllenZhu et. al. '16, Peng et. al. '16, Wang et. al. '15). For a wide variety of structured instances commonly found in applications, the iterations of our algorithm run in nearlylinear time. In doing so, we give matrix analytic techniques for overcoming obstacles that have stymied prior approaches to this open problem, as stated in past works (Peng et. al. '16, Mahoney et. al. '16). Crucial to our analysis are a simplification of existing algorithms for mixed positive linear programs, achieved by removing an asymmetry caused by modifying covering constraints, and a suite of matrix inequalities whose proofs are based on analyzing the Schur complements of matrices in a higher dimension. We hope that both our algorithm and techniques open the door to improved solvers for positive semidefinite programming, as well as its applications.
 Publication:

arXiv eprints
 Pub Date:
 February 2020
 arXiv:
 arXiv:2002.04830
 Bibcode:
 2020arXiv200204830J
 Keywords:

 Computer Science  Data Structures and Algorithms;
 Mathematics  Optimization and Control
 EPrint:
 51 pages, STOC 2020