# Using Optimization to Obtain a Width-Independent, Parallel, Simpler, and Faster Positive SDP Solver

### Abstract

We study the design of polylogarithmic depth algorithms for approximately solving packing and covering semidefinite programs (or positive SDPs for short). This is a natural SDP generalization of the well-studied positive LP problem. Although positive LPs can be solved in polylogarithmic depth while using only $\log^2 n/ \epsilon^3$ parallelizable iterations [4], the best known positive SDP solvers due to Jain and Yao [18] require $\log^14 n/ \epsilon^13$ parallelizable iterations. Several alternative solvers have been proposed to reduce the exponents in the number of iterations [19, 30]. However, the correctness of the convergence analyses in these works has been called into question [30], as they both rely on algebraic monotonicity properties that do not generalize to matrix algebra. In this paper, we propose a very simple algorithm based on the optimization framework proposed in [4] for LP solvers. Our algorithm only needs $\log^2 n/ \epsilon^3$ iterations, matching that of the best LP solver. To surmount the obstacles encountered by previous approaches, our analysis requires a new matrix inequality that extends Lieb-Thirring’s inequality, and a sign-consistent, randomized variant of the gradient truncation technique proposed in [3, 4].

Type
Publication
Proceedings of the 27th ACM-SIAM Symposium on Discrete Algorithms