Convergence Rates with Inexact Non-expansive Operators

In this paper, we present a convergence rate analysis for the inexact Krasnosel'skii-Mann iteration built from nonexpansive operators. Our results include two main parts: we first establish global pointwise and ergodic iteration-complexity bounds, and then, under a metric subregularity assumption, we establish local linear convergence for the distance of the iterates to the set of fixed points. The obtained iteration-complexity result can be applied to analyze the convergence rate of various monotone operator splitting methods in the literature, including the Forward-Backward, the Generalized Forward-Backward, Douglas-Rachford, alternating direction method of multipliers (ADMM) and Primal-Dual splitting methods. For these methods, we also develop easily verifiable termination criteria for finding an approximate solution, which can be seen as a generalization of the termination criterion for the classical gradient descent method. We finally develop a parallel analysis for the non-stationary Krasnosel'skii-Mann iteration. The usefulness of our results is illustrated by applying them to a large class of structured monotone inclusion and convex optimization problems. Experiments on some large scale inverse problems in signal and image processing problems are shown.Comment: This is an extended version of the work presented in http://arxiv.org/abs/1310.6636, and is accepted by the Mathematical Programmin

GMRES-Accelerated ADMM for Quadratic Objectives

We consider the sequence acceleration problem for the alternating direction method-of-multipliers (ADMM) applied to a class of equality-constrained problems with strongly convex quadratic objectives, which frequently arise as the Newton subproblem of interior-point methods. Within this context, the ADMM update equations are linear, the iterates are confined within a Krylov subspace, and the General Minimum RESidual (GMRES) algorithm is optimal in its ability to accelerate convergence. The basic ADMM method solves a $\kappa$-conditioned problem in $O(\sqrt{\kappa})$ iterations. We give theoretical justification and numerical evidence that the GMRES-accelerated variant consistently solves the same problem in $O(\kappa^{1/4})$ iterations for an order-of-magnitude reduction in iterations, despite a worst-case bound of $O(\sqrt{\kappa})$ iterations. The method is shown to be competitive against standard preconditioned Krylov subspace methods for saddle-point problems. The method is embedded within SeDuMi, a popular open-source solver for conic optimization written in MATLAB, and used to solve many large-scale semidefinite programs with error that decreases like $O(1/k^{2})$, instead of $O(1/k)$, where $k$ is the iteration index.Comment: 31 pages, 7 figures. Accepted for publication in SIAM Journal on Optimization (SIOPT