(558f) A Nested Schur Decomposition Approach for Multiperiod Optimization of Chemical Process

This work develops an algorithm for solving multiperiod optimization (MPO) problems using a nested Schur decomposition (NSD) approach. MPOs are an important class of problems that usually consider design and operation plans for predicted demands over a given time horizon. MPOs are also one of multi-scale integration problems from a single process model to MPOs related to business decisions. For chemical processes, rigorous process models are highly integrated with multiple unit models and thermodynamic property models, which lead to a large system of nonlinear and non-convex equations. MPOs of such process models could be significantly larger and more complex than a single process model. In the current volatile market situation, the chemical industry highly demands strategies to transit multi-scale problems in a robust, flexible, and efficient way over their process development. Therefore, we develop and demonstrate the NSD approach with an MPO of an industrial process model in this work.

The NSD approach decomposes MPO using a Schur complement, and allows us to solve the decomposed problem in parallel. Unlike other internal decomposition approaches (e.g., Schur-IPOPT)[1], the NSD partitions the problem into a two-level problem at the problem level. The problem level decomposition could be easier and more flexible than the internal decomposition methods. The two-level problem consists of the upper and lower problems. In MPOs, the upper problem has inventory, demand, and design constraints set over the entire period. On the other hand, the lower problem consists of a single process model for each period. There are, however, complicating variables that need to exist in both upper and lower problems, such as production rates. Hence, each lower problem is disaggregated by introducing artificial variables and constraints, which force the values of the artificial variables onto the complicating variables in the lower problem. It is noted that each lower problem could be infeasible or overdetermined by strictly forcing the constraints for complicating variables such as production rates. Thus, those constraints are relaxed, and its violations are penalized in the lower problem of the NSD.

For the solution strategy, the NSD converges the disaggregated problem in each period as the lower level problem, and the complicating variables are solved in the upper level problem. The detailed solution strategy is as follows. First, the lower problem is solved for each period and collects the KKT (Karush Kuhn Tucker) matrices. Then, the reduced Hessian matrix for the complicating variables is constructed from the assembled KKT system using the Schur complement. The reduced gradient vector for the complicating variables is obtained with sensitivity information of each period; this step can be parallelized. Finally, the upper problem updates the complicating variables with the Hessian and gradient information by using a globally convergent Newton-type solver. In the implementation, Pyomo is used to define both upper and lower problems. The KKT system is retrieved from the lower problem solution with Pynumero after solving the lower problem with IPOPT or CONOPT. The lower problem solver can be selected based on the inner problem condition. For example, IPOPT is efficient when the problem is well-conditioned. CONOPT is robust if the problem is ill-conditioned. Then, the Hessian and gradient information obtained by Schur complement are augmented into the upper problem. The augmented upper problem is solved with IPOPT iteratively.

An industrial benzene chlorination (BC) process for the MPO is used to demonstrate the NSD approach in this work. The BC process model contains approximately 10,000 constraints and 30 degrees of freedom. The lower problem consists of the BC process model, and the production rates are considered the complicating variables. In the upper problem, the problem is formulated with inventory, demand constraints, and production rates. The entire MPO problem will be at least 10,000 × N constraints and 30 × N degrees of freedom, where N is the number of periods. For solving the lower problem, CONOPT4 is used, which is one of the most robust active set solvers for nonlinear optimization. The NSD solution is compared with the direct solution, where the entire MPO is simultaneously solved without the decomposition by CONOPT4. From the result, the NSD successfully converges to the same optimum of the direct solution. The parallel computation of the NSD shows a significant time reduction in computation. The computational time of the parallel NSD increases sublinearly while the direct solution linearly increases in terms of . Thus, the parallel NSD has the advantage for large-scale MPO.

In summary, NSD is developed and demonstrated with an industrial multiperiod process optimization problem. The method allows us flexibly to choose the inner problem solver. Especially, we have demonstrated the NSD with CONOPT4 in this work. In addition, the NSD has a similar relationship between IPOPT and CONOPT, which are infeasible and feasible path solvers, respectively, in contrast to Schur-IPOPT. The NSD enforces feasibility in the inner problem over the iterations. This feature could be the reason for the robustness. Furthermore, the parallel NSD could outperform the computational time over the direct solution approach in the MPO with a large number of periods. Therefore, NSD provides a robust, flexible, and efficient option for industry to solve large MPOs.

References

[1] Zavala VM, Laird CD, Biegler LT. Interior-point decomposition approaches for parallel solution of large-scale nonlinear parameter estimation problems. Chem Eng Sci. 2008;63(19):4834-4845