The alternating direction method of multipliers (ADMM) is widely used in solving structured convex optimization problems. Despite of its success in practice, the convergence of the standard ADMM for minimizing the sum of $N$ $(N\geq 3)$ convex functions whose variables are linked by linear constraints, has remained unclear for a very long time. Recently, Chen … Read more
The alternating direction method of multipliers (ADMM) is a benchmark for solving a linearly constrained convex minimization model with a two-block separable objective function; and it has been shown that its direct extension to a multiple-block case where the objective function is the sum of more than two functions is not necessarily convergent. For the … Read more
Recently, a strictly contractive Peaceman- Rachford splitting method (SC-PRSM) was proposed to solve a convex minimization model with linear constraints and a separable objective function which is the sum of two functions without coupled variables. We show by an example that the SC-PRSM cannot be directly extended to the case where the objective function is … Read more
In this paper, we present a convergence rate analysis for the inexact Krasnosel’ski{\u{\i}}-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 … Read more
When the alternating direction method of multipliers (ADMM) is extended directly to a multi-block separable convex minimization model whose objective function is in form of more than two functions without coupled variables, it was recently shown that the convergence is not guaranteed. This fact urges to develop efficient algorithms that can preserve completely the numerical … Read more
This paper introduces a parallel and distributed extension to the alternating direction method of multipliers (ADMM). The algorithm decomposes the original problem into N smaller subproblems and solves them in parallel at each iteration. This Jacobian-type algorithm is well suited for distributed computing and is particularly attractive for solving certain large-scale problems. This paper introduces … Read more
Recently several methods were proposed for sparse optimization which make careful use of second-order information [11, 30, 17, 3] to improve local convergence rates. These methods construct a composite quadratic approximation using Hessian information, optimize this approximation using a first-order method, such as coordinate descent and employ a line search to ensure sufficient descent. Here … Read more
In this paper, we analyze the iteration-complexity of a generalized forward-backward (GFB) splitting algorithm, recently proposed in~\cite{gfb2011}, for minimizing the large class of composite objectives $f + \sum_{i=1}^n h_i$ on a Hilbert space, where $f$ has a Lipschitz-continuous gradient and the $h_i$’s are simple (i.e. whose proximity operator is easily computable ). We derive iteration-complexity … Read more
We propose a generalized proximal point algorithm (PPA), in the generic setting of finding a zero point of a maximal monotone operator. In addition to the classical PPA, a number of benchmark operator splitting methods in PDE and optimization literatures such as the Douglas-Rachford splitting method, Peaceman-Rachford splitting method, alternating direction method of multipliers, generalized … Read more
The alternating direction method of multipliers (ADMM) has emerged as a powerful technique for large-scale structured optimization. Despite many recent results on the convergence properties of ADMM, a quantitative characterization of the impact of the algorithm parameters on the convergence times of the method is still lacking. In this paper we find the optimal algorithm … Read more