MPEC Problem Formulations in Chemical Engineering Applications

7 ConclusionsWith the development and widespread use of large-scale nonlinear programmingtools for process optimization, there has been an associated applicationof complementarity formulations to represent discrete decisions. This study investigatesboth the formulation and solution strategies for the MathematicalPrograms with Compementarity Constraints (MPCCs). Because MPCCs havedependent constaints at all feasible points along with unbounded multipliers,special care is needed in the formulation of associated optimization problemsalong with a reliable solution algorithm. In this study, we investigate and summarizea number of regularization and penalty MPCC formulations and findthat the penalty formulation is particularly advantageous, especially when coupledwith an active set NLP solver. This conclusion is observed in a numericalcomparison with a library of MPCC test problems.Because all MPCCs can be formulated as underlying MPEC problems thatresult from nested optimization formulations, a necessary condition for wellbehavedMPCCs is that the inner optimization problem be convex with solutionsthat are not restricted by upper level constraints. This condition is used to motivatea systematic strategy for the formulation of complementarity constraints.This strategy is applied to a number of widely-used examples in process engineering,including flow reversals, relief valves, compressor kickbacks, controllersaturation and disappearance of phases in equilibrium stages.This combination of well-posed formulations and MPCC solutions strategiesis demonstrated on two large-scale case studies with as many as 8000 discretedecisions. First, in the discretized dynamic system with switches, both thepenalty and relaxed formulations are very effective with active set and barriersolvers, respectively. Second, in dealing with disappearing phases in distillationoptimization models, the penalty formulation shows significant performance improvementsover the NCP formulation in a previous study. From these results,we find that well-posed complementarities coupled with NLP problems basedon penalty formulations are efficient and effective strategies for the solution ofMPCCs in process engineering.AcknowledgementsFunding from the National Science Foundation under grant #CTS-0438279 isgratefully appreciated. The authors also wish to thank Dr. Steven Dirkse forfruitful discussions and help with the NLPEC package.References[1] Anitescu, M., P. Tseng, and S.J. Wright, “Elastic-mode algorithms formathematical programs with equilibrium constraints: global convergenceand stationarity properties,” Math. Program., 110, pp. 337-371 (2007).26