A Subspace Shift Technique for Nonsymmetric Algebraic Riccati Equations

The worst situation in computing the minimal nonnegative solution of a nonsymmetric algebraic Riccati equation associated with an M-matrix occurs when the corresponding linearizing matrix has two very small eigenvalues, one with positive and one with negative real part. When both these eigenvalues are exactly zero, the problem is called critical or null recurrent. While in this case the problem is ill-conditioned and the convergence of the algorithms based on matrix iterations is slow, there exist some techniques to remove the singularity and transform the problem to a well-behaved one. Ill-conditioning and slow convergence appear also in close-to-critical problems, but when none of the eigenvalues is exactly zero the techniques used for the critical case cannot be applied. In this paper, we introduce a new method to accelerate the convergence properties of the iterations also in close-to-critical cases, by working on the invariant subspace associated with the problematic eigenvalues as a whole. We present a theoretical analysis and several numerical experiments which confirm the efficiency of the new method

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

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 $\tilde{O}(\log^{2} n/\varepsilon^2)$ parallelizable iterations, the best known positive SDP solvers due to Jain and Yao require $O(\log^{14} n /\varepsilon^{13})$ parallelizable iterations. Several alternative solvers have been proposed to reduce the exponents in the number of iterations. However, the correctness of the convergence analyses in these works has been called into question, 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 for LP solvers. Our algorithm only needs $\tilde{O}(\log^2 n / \varepsilon^2)$ 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