(522f) Analysis of the Alternating Direction Method of Multipliers for the Optimization of Distributed Nonconvex Systems

The alternating direction method of multipliers (ADMM) is an optimization method that has received a lot of attention for solving convex optimization problems. Some of this attention is because ADMM can be parallelized for problems with a certain separable structure. It is particularly suited to problems that have a decentralized or distributed nature.

However, very little analysis has been done on the theoretical properties and numerical performance of ADMM for the nonconvex problems of interest in chemical engineering applications. As a motivating example, consider the coordination or integration of various real-time optimization (RTO) applications across a refinery. Each RTO application can be a large nonconvex optimization problem with the aim of achieving economically optimal steady-state operation of a particular process unit or group of units. Of course these units, with separate RTO applications, are actually connected by material flows. Thus, the RTO problems can be connected by extra constraints in order to obtain a larger-scope optimization problem. Ideally, an optimization method for this larger, integrated optimization problem could take advantage of the existing infrastructure, and “sit on top” of the models and software in use for the RTO applications. At least superficially, ADMM is a promising candidate, and numerical experiments with problems with similar structure show that ADMM does seem to work.

This work aims to support these numerical experiments and provide a convergence analysis of ADMM applied to nonconvex problems. Under fairly standard regularity assumptions on a local minimizer, we establish conditions under which ADMM does indeed converge to that minimum. These conditions include that the penalty parameter in the augmented Lagrangian be greater than a certain threshold. Bolstered by these theoretical guarantees, we propose putting ADMM in a trust region framework to improve its robustness and global convergence properties.