English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

In this paper, we analyze the convergence of Alternating Direction Method ofMultipliers (ADMM) on convex quadratic programs (QPs) with linear equality andbound constraints. The ADMM formulation alternates between an equalityconstrained QP and a projection on the bounds. Under the assumptions of (i)positive definiteness of the Hessian of the objective projected on the nullspace of equality constraints (reduced Hessian), and (ii) linear independenceconstraint qualification holding at the optimal solution, we derive an upperbound on the rate of convergence to the solution at each iteration. Inparticular, we provide an explicit characterization of the rate of convergencein terms of: (a) the eigenvalues of the reduced Hessian, (b) the cosine of theFriedrichs angle between the subspace spanned by equality constraints and thesubspace spanned by the gradients of the components that are active at thesolution and (c) the distance of the inactive components of solution from thebounds. Using this analysis we show that if the QP is feasible, the iteratesconverge at a Q-linear rate and prescribe an optimal setting for the ADMMstep-size parameter. For infeasible QPs, we show that the primal variables inADMM converge to minimizers of the Euclidean distance between the hyperplanedefined by the equality constraints and the convex set defined by the bounds.The multipliers for the bound constraints are shown to diverge along the rangespace of the equality constraints. Using this characterization, we also proposea termination criterion for ADMM. Numerical examples are provided to illustratethe theory through experiments.

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