A Partially Inexact Alternating Direction Method of Multipliers and its Iteration-Complexity Analysis

This paper proposes a partially inexact alternating direction method of multipliers for computing approximate solution of a linearly constrained convex optimization problem. This method allows its first subproblem to be solved inexactly using a relative approximate criterion, whereas a proximal term is added to its second subproblem in order to simplify it. A stepsize parameter is included in the updating rule of the Lagrangian multiplier to improve its computational performance. Pointwise and ergodic interation-complexity bounds for the proposed method are established. To the best of our knowledge, this is the first time that complexity results for an inexact ADMM with relative error criteria has been analyzed. Some preliminary numerical experiments are reported to illustrate the advantages of the new method

An inexact version of the symmetric proximal ADMM for solving separable convex optimization

In this paper, we propose and analyze an inexact version of the symmetric proximal alternating direction method of multipliers (ADMM) for solving linearly constrained optimization problems. Basically, the method allows its first subproblem to be solved inexactly in such way that a relative approximate criterion is satisfied. In terms of the iteration number $k$, we establish global $\mathcal{O} (1/ \sqrt{k})$ pointwise and $\mathcal{O} (1/ {k})$ ergodic convergence rates of the method for a domain of the acceleration parameters, which is consistent with the largest known one in the exact case. Since the symmetric proximal ADMM can be seen as a class of ADMM variants, the new algorithm as well as its convergence rates generalize, in particular, many others in the literature. Numerical experiments illustrating the practical advantages of the method are reported. To the best of our knowledge, this work is the first one to study an inexact version of the symmetric proximal ADMM

Fast and Stable Nonconvex Constrained Distributed Optimization: The ELLADA Algorithm

Distributed optimization, where the computations are performed in a localized and coordinated manner using multiple agents, is a promising approach for solving large-scale optimization problems, e.g., those arising in model predictive control (MPC) of large-scale plants. However, a distributed optimization algorithm that is computationally efficient, globally convergent, amenable to nonconvex constraints and general inter-subsystem interactions remains an open problem. In this paper, we combine three important modifications to the classical alternating direction method of multipliers (ADMM) for distributed optimization. Specifically, (i) an extra-layer architecture is adopted to accommodate nonconvexity and handle inequality constraints, (ii) equality-constrained nonlinear programming (NLP) problems are allowed to be solved approximately, and (iii) a modified Anderson acceleration is employed for reducing the number of iterations. Theoretical convergence towards stationary solutions and computational complexity of the proposed algorithm, named ELLADA, is established. Its application to distributed nonlinear MPC is also described and illustrated through a benchmark process system.Comment: 18 pages, 5 figure