(749g) Accelerated Parallel Alternating Method of Multipliers (ADMM) for Distributed Optimization

Authors: 
Tang, W., University of Minnesota, Twin Cities
Daoutidis, P., University of Minnesota, Twin Cities
The real-time optimization in the operation of distributed large-scale plants demands effective distributed optimization algorithms, where the optimization problems are solved based on a system decomposition. That is, the variables in the entire optimization problem are partitioned into several blocks with interconnecting constraints, and the resulting sub-problems are optimized in a distributed manner, coordinated through the information change between them.

Alternating direction method of multipliers (ADMM) is a widely used algorithm of distributed optimization, especially for problems in machine learning and image processing [1]. For chemical engineering, its application to distributed real-time optimization has been recently introduced [2]. ADMM can also be used for distributed model predictive control (see e.g. [3]). A significant feature of ADMM is that it allows each optimizer (controller) to work with only local information, i.e. the model of the corresponding subsystem, its connections with upstream and downstream variables, and its own optimization objective and constraints. In this sense, ADMM is a localized distributed optimization algorithm suitable for local information availability.

However, the real-time implementation of the classical multi-block ADMM algorithm is largely prohibited by a high computational cost due to its linear convergence rate and the sequential optimization scheme of solving the sub-problems. Recently, there have been works that seek to parallelize the multi-block optimization in ADMM (see e.g. [4]), and accelerate the convergence of two-block ADMM to a quadratic rate (see e.g. [5]).

In this work, we incorporate the efforts of [4] and [5] to parallelize and accelerate ADMM in a novel accelerated parallel ADMM algorithm in the context of solving multi-block, nonconvex, constrained optimization problems, which arise in the distributed control or real-time optimization of nonlinear chemical process systems. We demonstrate with a case study that the ADMM formulation can be applied to the localized distributed control of a chemical plant, and that the accelerated parallel ADMM significantly improves the computational performance compared to the classical ADMM.

References

[1] Boyd, S., et al. (2011). Distributed optimization and statistical learning via the alternating direction method of multipliers. Found. Trends Mach. Learn., 3(1), 1-122.

[2] Harwood, S., et al. (2017). Analysis of the alternating direction method of multipliers for the optimization of distributed nonconvex systems. 2017 AIChE Annual Meeting, 522f.

[3] Mota, J. F., et al. (2015). Distributed optimization with local domains: Applications in MPC and network flows. IEEE Trans. Autom. Control, 60(7), 2004-2009.

[4] Deng, W., et al. (2017). Parallel multi-block ADMM with o(1/k) convergence. J. Sci. Comput., 71(2), 712-736.

[5] Goldstein, T., et al. (2014). Fast alternating direction optimization methods. SIAM J. Imaging Sci., 7(3), 1588-1623.

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