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Randomized Primal–Dual Proximal Block Coordinate Updates

Journal of the Operations Research Society of China 2019年第2期7卷 205-250页

作者： Xiang Gao Yang-Yang Xu Shu-Zhong Zhang Department of Industrial and Systems Engineering University of MinnesotaMinneapolisUSA Department of Mathematical Sciences Rensselaer Polytechnic InstituteTroyUSA

In this paper,we propose a randomized primal–dual proximal block coordinate updating framework for a general multi-block convex optimization model with coupled objective function and linear *** mere convexity,we establish its O(1/t)convergence rate in terms of the objective value and feasibility *** framework includes several existing algorithms as special cases such as a primal–dual method for bilinear saddle-point problems(PD-S),the proximal Jacobian alternating direction method of multipliers(Prox-JADMM)and a randomized variant of the ADMM for multi-block convex *** analysis recovers and/or strengthens the convergence properties of several existing *** example,for PD-S our result leads to the same order of convergence rate without the previously assumed boundedness condition on the constraint sets,and for Prox-JADMM the new result provides convergence rate in terms of the objective value and the feasibility *** is well known that the original ADMM may fail to converge when the number of blocks exceeds *** result shows that if an appropriate randomization procedure is invoked to select the updating blocks,then a sublinear rate of convergence in expectation can be guaranteed for multi-block ADMM,without assuming any strong *** new approach is also extended to solve problems where only a stochastic approximation of the subgradient of the objective is available,and we establish an O(1/√t)convergence rate of the extended approach for solving stochastic programming.

关键词： Primal-dual method Alternating direction method of multipliers(ADMM) Randomized algorithm iteration complexity·first-order stochastic approximation

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