73 research outputs found
A semi-proximal-based strictly contractive Peaceman-Rachford splitting method
The Peaceman-Rachford splitting method is very efficient for minimizing sum
of two functions each depends on its variable, and the constraint is a linear
equality. However, its convergence was not guaranteed without extra
requirements. Very recently, He et al. (SIAM J. Optim. 24: 1011 - 1040, 2014)
proved the convergence of a strictly contractive Peaceman-Rachford splitting
method by employing a suitable underdetermined relaxation factor. In this
paper, we further extend the so-called strictly contractive Peaceman-Rachford
splitting method by using two different relaxation factors, and to make the
method more flexible, we introduce semi-proximal terms to the subproblems. We
characterize the relation of these two factors, and show that one factor is
always underdetermined while the other one is allowed to be larger than 1. Such
a flexible conditions makes it possible to cover the Glowinski's ADMM whith
larger stepsize. We show that the proposed modified strictly contractive
Peaceman-Rachford splitting method is convergent and also prove
convergence rate in ergodic and nonergodic sense, respectively. The numerical
tests on an extensive collection of problems demonstrate the efficiency of the
proposed method
Scalable Peaceman-Rachford Splitting Method with Proximal Terms
Along with developing of Peaceman-Rachford Splittling Method (PRSM), many
batch algorithms based on it have been studied very deeply. But almost no
algorithm focused on the performance of stochastic version of PRSM. In this
paper, we propose a new stochastic algorithm based on PRSM, prove its
convergence rate in ergodic sense, and test its performance on both artificial
and real data. We show that our proposed algorithm, Stochastic Scalable PRSM
(SS-PRSM), enjoys the convergence rate, which is the same as those
newest stochastic algorithms that based on ADMM but faster than general
Stochastic ADMM (which is ). Our algorithm also owns wide
flexibility, outperforms many state-of-the-art stochastic algorithms coming
from ADMM, and has low memory cost in large-scale splitting optimization
problems
Tight Global Linear Convergence Rate Bounds for Douglas-Rachford Splitting
Recently, several authors have shown local and global convergence rate
results for Douglas-Rachford splitting under strong monotonicity, Lipschitz
continuity, and cocoercivity assumptions. Most of these focus on the convex
optimization setting. In the more general monotone inclusion setting, Lions and
Mercier showed a linear convergence rate bound under the assumption that one of
the two operators is strongly monotone and Lipschitz continuous. We show that
this bound is not tight, meaning that no problem from the considered class
converges exactly with that rate. In this paper, we present tight global linear
convergence rate bounds for that class of problems. We also provide tight
linear convergence rate bounds under the assumptions that one of the operators
is strongly monotone and cocoercive, and that one of the operators is strongly
monotone and the other is cocoercive. All our linear convergence results are
obtained by proving the stronger property that the Douglas-Rachford operator is
contractive
Linear Convergence and Metric Selection for Douglas-Rachford Splitting and ADMM
Recently, several convergence rate results for Douglas-Rachford splitting and
the alternating direction method of multipliers (ADMM) have been presented in
the literature. In this paper, we show global linear convergence rate bounds
for Douglas-Rachford splitting and ADMM under strong convexity and smoothness
assumptions. We further show that the rate bounds are tight for the class of
problems under consideration for all feasible algorithm parameters. For
problems that satisfy the assumptions, we show how to select step-size and
metric for the algorithm that optimize the derived convergence rate bounds. For
problems with a similar structure that do not satisfy the assumptions, we
present heuristic step-size and metric selection methods
A faster prediction-correction framework for solving convex optimization problems
He and Yuan's prediction-correction framework [SIAM J. Numer. Anal. 50:
700-709, 2012] is able to provide convergent algorithms for solving convex
optimization problems at a rate of in both ergodic and pointwise
senses. This paper presents a faster prediction-correction framework at a rate
of in the non-ergodic sense and in the pointwise sense,
{\it without any additional assumptions}. Interestingly, it provides a faster
algorithm for solving {\it multi-block} separable convex optimization problems
with linear equality or inequality constraints
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