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We consider a family of parallel methods for constrained optimization basedon projected gradient descents along individual coordinate directions. In thecase of polyhedral feasible sets, local convergence towards a regular solutionoccurs unconstrained in a reduced space, allowing for the computation of tightasymptotic convergence rates by sensitivity analysis, this even when globalconvergence rates are unavailable or too conservative. We derive linearasymptotic rates of convergence in polyhedra for variants of the coordinatedescent approach, including cyclic, synchronous, and random modes ofimplementation. Our results find application in stochastic optimization, andwith recently proposed optimization algorithms based on Taylor approximationsof the Newton step.

Asymptotic convergence rates for coordinate descent in polyhedral sets