Mathematical programs with equilibrium constraints (MPECs) in process engineering

Mathematical programming approaches have proved invaluable in the design and operation of chemical processes. This has been enabled through the development of improved modeling constructs for the accurate representation of complex phenomena and the development of efficient algorithms that can solve the new modeling constructs. This dissertation is concerned with the use of one such construct, complementarity constraints and variational inequalities (VIs), for the modeling of process engineering applications.; Complementarity constraints and VIs can be shown to model a number of discrete and discontinuous behavior that is typical of most engineering systems. Optimization problems constrained by complementarity constraints and VIs are commonly referred to as Mathematical Programs with Equilibrium Constraints (MPECs). On the other hand, MPECs do not satisfy regularity assumptions that are commonly assumed for nonlinear programs and hence, the solution of MPECs pose a huge challenge. This dissertation aims to (i) develop an algorithm for the solution of MPECs, (ii) identify applications in process engineering that can be cast as an MPEC and (iii) demonstrate the performance of algorithm on applications.; Assumptions of convexity and regularity on variational inequalities allow their reformulation to complementarity constraints. Solution of the MPEC is considered by posing as a Mathematical Program with Complementarity Constraints (MPCC). An interior point approach is proposed for the solution of MPCCs, thereby avoiding the combinatorial complexity of identifying the active constraints. The proposed approach adapts the interior point methods for nonlinear programs (NLPs) for the solution of MPCCs. The complementarity constraints of the NIPCC are suitably relaxed so as to guarantee a strictly feasible region for the inequality constraints as required by the interior point approach. The relaxation is related to the barrier parameter and is driven to zero in the limit. The primal-dual step calculation is modified to handle the singularity in the limit. The modification is shown to allow for fast local convergence of the algorithm. The proposed step modification needs to be performed only close to a solution. Thus, existing NLP interior point algorithms can be easily adapted for this purpose. The practical performance of the proposed method has been verified by incorporating the modification within one such NLP interior point algorithm. Encouraging numerical performance on a large number of MPEC test problems has been observed.; Equilibrium constraints are shown to arise in the modeling of distillation columns with phase changes and in biology. Steady state design of a distillation column and dynamic optimization of a binary batch distillation column and cryogenic distillation column are presented. Data reconciliation and parameter estimation in metabolic flux balance models is posed as an MPEC and solved reliably. Finally, the parameter estimation of a dynamic batch fermentation is modeled as a Differential Variational Inequality (DVI). Discretization of the DVI yields an NIPEC. Encouraging numerical results are reported.