Higher operating costs and shrinking profit margins in the chemical and petro-chemical industries is driving greater applications of advanced control techniques, as well as consideration of control at the process design stage. These applications are often model-based and require the solution of optimization formulations comprising very large systems of variables and equations that can be computationally demanding, requiring considerable computer memory and solution times. Accordingly, these applications are motivating the development and implementation of solution approaches, numerical techniques and algorithms capable of efficiently exploiting modern computational resources in terms utilizing multiprocessor systems and/or acceleration devices, all while limiting memory usage. One particular direction that we focus on in this work is a solution strategy for multiperiod dynamic optimization formulations which incorporates process models described by differential-algebraic equations (DAEs) with uncertain parameters and/or disturbance inputs. To this end, we are investigating a direct multiperiod discretization approach of the uncertainty space combined with a multiple-shooting discretization of the temporal domain, which requires the solution of an embedded dynamic model within an overall nonlinear programming algorithm.
Over the last few decades interior-point optimization approaches have been shown to be superior to other methods in terms of efficiently handling ever increasing large-scale optimization formulations. For stochastic optimization problems, multiperiod (or multiscenario) discretization is often applied where numerous scenario realizations are generated by sampling from a particular distribution or within a defined interval. The resulting formulations are posed using a sample-average objective function and constraints for all realizations . Accordingly, this introduces a partially separable block structure with design variables representing coupling or complicating variables between each scenario realization. These particular formulations have led to the development of structure exploiting solution strategies that utilize efficient decomposition approaches, which either aim to reduce the dimensionality of the original formulation allowing standard dense linear algebra techniques or exploit the separability of the full-space structure allowing a portion of the algorithm to be simultaneously solved in parallel. In this latter direction, one such approach has been the determination of the Newton search direction within linear, quadratic or nonlinear programs via a Schur-complement decomposition where certain aspects of approach can be performed in parallel while the Schur-complement itself can be formed and then factored directly  or perhaps more efficiently, for very large linear systems with numerous complicating variables, solved iteratively using preconditioned conjugate gradient methods .
Our proposed solution approach extends our previous work  and uses a sequential-quadratic programming (SQP) algorithm where the QP subproblems are solved using an interior-point method (IPM) via the OOPS solver which employs a direct Schur-complement decomposition . Additionally, a partially separable quasi-Newton strategy is used to iteratively approximate and update second-order information used in formulating the QP. Novel aspects of the work include parallelization of the DAE solution through a multiple shooting approach, coupled with parallelization of the uncertainty scenarios, and employment of an SQP-IPM large-scale nonlinear programming algorithm that exploits the structure of discretized dynamic optimization formulations via decomposition.
The focus of our presentation will be on describing our proposed strategy for efficiently decomposing and solving large-scale multiperiod dynamic optimization formulations. More specifically, we will highlight the following aspects: (1) our approach for decomposing and parallelizing embedded DAE model representations within a multiple-shooting dynamic optimization framework; (2) an assessment of the potential computation improvements of the parallel algorithm for solving embedded DAEs utilizing an adaptive DAE solver with simultaneous forward sensitivity generation; (3) an assessment of our SQP-IPM nonlinear programming approach which utilizes a structure exploiting QP solver. Our implementation makes use of a c/c++ computing environment that uses (1) OpenMP loop parallelization constructs to evaluate the embedded DAE model via the Sundials IDAS solver; (2) the QP solver OOPS for each subproblem which has the potential to be parallelized using a distributed computing MPI implementation. Case studies involving integrated design and control and/or robust control will be presented and numerical results investigated, which aim to demonstrate the potential computing speedup and efficiency of the parallelized multiperiod algorithm with increasing processors and load. Additionally, avenues for future research will be identified and some perspective provided on the application of the proposed techniques to industrial large-scale DAE models.
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