Activity Identification and Local Linear Convergence of Douglas--Rachford/ADMM under Partial Smoothness

Liang, Jingwei; Fadili, Jalal M.; Peyré, Gabriel; Luke, Russell

doi:10.48550/arxiv.1412.6858

Cited by 1 publication

(1 citation statement)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Liang et al [31] This paper was published after our initial submission but we include this for completeness. Liang et al [31] characterize the finite active set identification and local linear convergence for the DR. We interpret the main assumptions in that paper in the context of the QP (1) as: (a) strict complementarity (eqn (3.1) in [31]) and (b) LICQ which is required to guarantee than the angle between the tangent spaces is bounded away from 0. No assumptions on the curvature of the Hessian of the objective is made.…”

Section: • In Our Notation Mmentioning

confidence: 99%

ADMM for Convex Quadratic Programs: Q-Linear Convergence and Infeasibility Detection

Raghunathan¹,

Cairano²

2014

Preprint

View full text Add to dashboard Cite

In this paper, we analyze the convergence of Alternating Direction Method of Multipliers (ADMM) on convex quadratic programs (QPs) with linear equality and bound constraints. The ADMM formulation alternates between an equality constrained QP and a projection on the bounds. Under the assumptions of (i) positive definiteness of the Hessian of the objective projected on the null space of equality constraints (reduced Hessian), and (ii) linear independence constraint qualification holding at the optimal solution, we derive an upper bound on the rate of convergence to the solution at each iteration. In particular, we provide an explicit characterization of the rate of convergence in terms of: (a) the eigenvalues of the reduced Hessian, (b) the cosine of the Friedrichs angle between the subspace spanned by equality constraints and the subspace spanned by the gradients of the components that are active at the solution and (c) the distance of the inactive components of solution from the bounds. Using this analysis we show that if the QP is feasible, the iterates converge at a Q-linear rate and prescribe an optimal setting for the ADMM step-size parameter. For infeasible QPs, we show that the primal variables in ADMM converge to minimizers of the Euclidean distance between the hyperplane defined by the equality constraints and the convex set defined by the bounds. The multipliers for the bound constraints are shown to diverge along the range space of the equality constraints. Using this characterization, we also propose a termination criterion for ADMM. Numerical examples are provided to illustrate the theory through experiments.

show abstract

Section: • In Our Notation Mmentioning

confidence: 99%