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Dynamic Optimization with Complementarity Constraints: Regularization for Direct Shooting

Adrian Caspari (adrian.caspari***at***avt.rwth-aachen.de)
Lukas Lüken (lukas.lueken***at***rwth-aachen.de)
Pascal Schäfer (pascal.schaefer***at***avt.rwth-aachen.de)
Yannic Vaupel (yannic.vaupel***at***avt.rwth-aachen.de)
Adel Mhamdi (adel.mhamdi***at***avt.rwth-aachen.de)
Lorenz T. Biegler (biegler***at***cmu.edu)
Alexander Mitsos (amitsos***at***alum.mit.edu)

Abstract: We consider the optimization of differential-algebraic equations (DAEs) with complementarity constraints (CCs) of algebraic state pairs. We formulate the CCs as nonlinear complementarity problem (NCP) functions. We regularize the NCP functions to obtain a smooth DAE, allowing for the solution via standard DAE integrators and NLP solvers in direct single-shooting. We provide a condition under which the original nonsmooth DAE is well-posed and show that these conditions are sufficient also for the regularized DAE to be well-posed. Thus, existing properties for algebraic optimization problems with CCs imply that with the regularization parameter going to zero, the solution of the optimization problem with regularized DAE converges to the solution of the original optimization problem. We present four case-studies: (i) optimal loading of an overflow weir buffer tank, (ii) batch vaporization setpoint tracking, (iii) operation of a tank cascade, and (iv) optimal start-up of a rectification column. The numerical results suggest that the presented approach scales favorably. We examine the required computational time for solution of the tank cascade problem for different number of tanks and compare the results to alternative solution methods. We demonstrate by example that the computational times of the solution approach scale not worse than quadratically with the problem size and do not scale with the control grid size.

Keywords: MPCCs with DAE, direct single-shooting, well-posedness analysis, optimization of regularized nonsmooth DAE

Category 1: Nonlinear Optimization

Category 2: Nonlinear Optimization (Systems governed by Differential Equations Optimization )

Category 3: Complementarity and Variational Inequalities

Citation: unpublished, under review. October 2019. Process Systems Engineering (AVT.SVT), RWTH Aachen University, 52074 Aachen, Germany. JARA-CSD, 52056 Aachen, Germany. Energy Systems Engineering (IEK-10), Forschungszentrum Jülich, 52425 Jülich, Germany. Carnegie Mellon University, Department of Chemical Engineering, Pittsburgh, PA 15213, USA.

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Entry Submitted: 10/01/2019
Entry Accepted: 10/01/2019
Entry Last Modified: 10/01/2019

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