Dynamic Optimization with Complementarity Constraints: Regularization for Direct Shooting-Supplementary Material

We consider the optimization of differential-algebraic equations (DAEs) with complementarity 9 constraints (CCs) of algebraic state pairs. We formulate the CCs as nonlinear complementarity problem 10 (NCP) functions. We regularize the NCP functions to obtain a smooth DAE, allowing for the solution 11 via standard DAE integrators and NLP solvers in direct single-shooting. We provide a condition under 12 which the original nonsmooth DAE is well-posed and show that these conditions are sufficient also for 13 the regularized DAE to be well-posed. Thus, existing properties for algebraic optimization problems with 14 CCs imply that with the regularization parameter going to zero, the solution of the optimization problem 15 with regularized DAE converges to the solution of the original optimization problem. We present four 16 case-studies: (i) optimal loading of an overflow weir buffer tank, (ii) batch vaporization setpoint tracking, 17 (iii) operation of a tank cascade, and (iv) optimal start-up of a rectification column. The numerical 18 results suggest that the presented approach scales favorably. We examine the required computational 19 time for solution of the tank cascade problem for different number of tanks and compare the results to 20 alternative solution methods. We demonstrate by example that the computational times of the solution 21 approach scale not worse than quadratically with the problem size and do not scale with the control grid 22 size. 23 24