Smoothing methods for complementarity problems in process engineering

Complementarity is central to all constrained optimization problems. Howeve r, direct enforcement of complementarity conditions is difficult because of the inherent nondifferentiability associated with them. Here, a class of s moothing methods for solving the complementarity problem by using a continu ation algorithm to solve nonlinear equations is studied. The applicability of smoothing methods to approximate complicated nested derivative discontin uities is investigated using simple functions with a single smoothing param eter. In addition, an equation-based formulation for solving the phase equi librium problem with complementarity conditions is formulated. This approac h can model the appearance and disappearance of phases directly in phase eq uilibrium problems. Moreover it is shown how smoothing methods can be used to solve limiting distillation cases, such as dry and vaporless trays, mode led within an equation-based formulation.