Calculation of phase and chemical equilibrium represents a crucial phase in the modeling of many separation processes. For conditions of constant temperature and pressure, a necessary and sufficient condition for the true equilibrium solution is that (i) the total Gibbs free energy of the system be at its global minimum, or (ii) the minimum of the tangent plane distance function be nonnegative for all phase models used to represent the system, In this work, the goal is to obtain equilibrium solutions corresponding to a global minimum of the Gibbs free energy as efficiently as possible, for cases where the liquid phase or phases can be modeled by the NRTL, UNIQUAC, UNIFAC, Wilson, modified Wilson and ASOG equations. Vapor phases whose behavior can be described as ideal can also be handled. In achieving this goal, there are two distinct problems of relevance : (i) the minimization of the Gibbs free energy, denoted (G), and (ii) the minimization of the tangent plane distance function, or the tangent plane stability criterion, denoted (S). For all these activity coefficient models, GLOPEQ (GLobal OPtimization for the Phase and chemical EQuilibrium problem) can guarantee global solutions for problems (G) and (S), but a combined algorithm employs them in tandem, using (G) to generate candidate equilibrium solutions which can then be verified for thermodynamic stability by solving (S). Two key features of the combined algorithm are that (i) as much information as is possible is obtained from local searches, and (ii) it is preferable to verify a globally stable equilibrium solution using the tangent plane criterion, as this problem contains fewer variables than the minimization of the Gibbs free energy. Results for several examples are presented, and all but one of them are for the case of phase equilibrium, due to the paucity of examples for reacting systems that employ excess Gibbs free energy models.