A global optimization algorithm was developed for finding all singular points (minima, maxima and saddles) of an objective function by exploring the natural connectedness that exists between their singular points. The idea of following ridges and valleys using information gathered along the way was significantly enhanced by applying the arc length continuation method. The algorithm was applied to the Gibbs tangent plane stability test for multiphase liquid mixtures. The global optimization algorithm gives an efficient and robust scheme for locating all stationary points of the tangent plane distance function predicted by any thermodynamic model. Since it provides very good initial estimates for the liquid-liquid equilibrium calculations, it became an integral part of a combined phase equilibrium and stability algorithm. The combined algorithm is self-starting and significantly improves reliability and robustness of multiphase equilibrium calculations. It was successfully tested on a variety of problems and is applicable to any component's mixtures and any number of liquid phases. Solutions have been found for the entire phase diagram. (C) 2013 Elsevier B.V. All rights reserved.