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
The augmented Lagrangian method (ALM) is a benchmark for solving the convex minimization problem with linear constraints. We consider the special case where the objective is in form of the sum of m functions without coupled variables. For solving this separable convex programming model, it is usually required to decompose the ALM subproblem at each … Read more
In this paper we analyze the randomized block-coordinate descent (RBCD) methods for minimizing the sum of a smooth convex function and a block-separable convex function. In particular, we extend Nesterov’s technique (SIOPT 2012) for analyzing the RBCD method for minimizing a smooth convex function over a block-separable closed convex set to the aforementioned more general … Read more
We propose a new first-order augmented Lagrangian algorithm ALCC for solving convex conic programs of the form min{rho(x)+gamma(x): Ax-b in K, x in chi}, where rho and gamma are closed convex functions, and gamma has a Lipschitz continuous gradient, A is mxn real matrix, K is a closed convex cone, and chi is a “simple” … Read more