688 research outputs found
Local Convergence Properties of Douglas–Rachford and Alternating Direction Method of Multipliers
International audienceThe Douglas–Rachford and alternating direction method of multipliers are two proximal splitting algorithms designed to minimize the sum of two proper lower semi-continuous convex functions whose proximity operators are easy to compute. The goal of this work is to understand the local linear convergence behaviour of Douglas–Rachford (resp. alternating direction method of multipliers) when the involved functions (resp. their Legendre-Fenchel conjugates) are moreover partly smooth. More precisely, when the two functions (resp. their conjugates) are partly smooth relative to their respective smooth submanifolds, we show that Douglas–Rachford (resp. alternating direction method of multipliers) (i) identifies these manifolds in finite time; (ii) enters a local linear convergence regime. When both functions are locally polyhe-dral, we show that the optimal convergence radius is given in terms of the cosine of the Friedrichs angle between the tangent spaces of the identified submanifolds. Under polyhedrality of both functions, we also provide conditions sufficient for finite convergence. The obtained results are illustrated by several concrete examples and supported by numerical experiments
Douglas-Rachford Splitting: Complexity Estimates and Accelerated Variants
We propose a new approach for analyzing convergence of the Douglas-Rachford
splitting method for solving convex composite optimization problems. The
approach is based on a continuously differentiable function, the
Douglas-Rachford Envelope (DRE), whose stationary points correspond to the
solutions of the original (possibly nonsmooth) problem. By proving the
equivalence between the Douglas-Rachford splitting method and a scaled gradient
method applied to the DRE, results from smooth unconstrained optimization are
employed to analyze convergence properties of DRS, to tune the method and to
derive an accelerated version of it
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
Local Convergence of Proximal Splitting Methods for Rank Constrained Problems
We analyze the local convergence of proximal splitting algorithms to solve
optimization problems that are convex besides a rank constraint. For this, we
show conditions under which the proximal operator of a function involving the
rank constraint is locally identical to the proximal operator of its convex
envelope, hence implying local convergence. The conditions imply that the
non-convex algorithms locally converge to a solution whenever a convex
relaxation involving the convex envelope can be expected to solve the
non-convex problem.Comment: To be presented at the 56th IEEE Conference on Decision and Control,
Melbourne, Dec 201
A Partition-Based Implementation of the Relaxed ADMM for Distributed Convex Optimization over Lossy Networks
In this paper we propose a distributed implementation of the relaxed
Alternating Direction Method of Multipliers algorithm (R-ADMM) for optimization
of a separable convex cost function, whose terms are stored by a set of
interacting agents, one for each agent. Specifically the local cost stored by
each node is in general a function of both the state of the node and the states
of its neighbors, a framework that we refer to as `partition-based'
optimization. This framework presents a great flexibility and can be adapted to
a large number of different applications. We show that the partition-based
R-ADMM algorithm we introduce is linked to the relaxed Peaceman-Rachford
Splitting (R-PRS) operator which, historically, has been introduced in the
literature to find the zeros of sum of functions. Interestingly, making use of
non expansive operator theory, the proposed algorithm is shown to be provably
robust against random packet losses that might occur in the communication
between neighboring nodes. Finally, the effectiveness of the proposed algorithm
is confirmed by a set of compelling numerical simulations run over random
geometric graphs subject to i.i.d. random packet losses.Comment: Full version of the paper to be presented at Conference on Decision
and Control (CDC) 201
On the Global Linear Convergence of the ADMM with Multi-Block Variables
The alternating direction method of multipliers (ADMM) has been widely used
for solving structured convex optimization problems. In particular, the ADMM
can solve convex programs that minimize the sum of convex functions with
-block variables linked by some linear constraints. While the convergence of
the ADMM for was well established in the literature, it remained an open
problem for a long time whether or not the ADMM for is still
convergent. Recently, it was shown in [3] that without further conditions the
ADMM for may actually fail to converge. In this paper, we show that
under some easily verifiable and reasonable conditions the global linear
convergence of the ADMM when can still be assured, which is important
since the ADMM is a popular method for solving large scale multi-block
optimization models and is known to perform very well in practice even when
. Our study aims to offer an explanation for this phenomenon
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