Complementarity is central to all constrained optimization problems. However, direct enforcement of complementarity conditions is difficult because of the inherent nondifferentiability associated with them. Here, a class of smoothing methods for solving the complementarity problem by using a continuation algorithm to solve nonlinear equations is studied. The applicability of smoothing methods to approximate complicated nested derivative discontinuities is investigated using simple functions with a single smoothing parameter. In addition, an equation-based formulation for solving the phase equilibrium problem with complementarity conditions is formulated. This approach can model the appearance and disappearance of phases directly in phase equilibrium problems. Moreover it is shown how smoothing methods can be used to solve limiting distillation cases, such as dry and vaporless trays, modeled within an equation-based formulation.