Thermodynamic stability analysis requires that the Gibbs free energy of an equilibrium mixture must be at global minimum. Several techniques use this criterion to identify the equilibrium phase-state of a multi-component mixture. In this article, we present an analysis of the Gibbs energy minimization in the Compositional Space Parameterization (CSP) framework. Our parameterizations are based on primary configurations of multi-dimensional tie-simplexes in the compositional space. We show that once the maximum number of coexisting phases in the "base parameterization" is known, CSP yields the global minimum of Gibbs energy in multiphase multi-component mixtures with complex phase behaviors. We present several challenging examples to demonstrate the equivalence of conventional techniques (based on global minimization of Gibbs energy) and the tie-simplex methodology. (C) 2012 Elsevier B.V. All rights resented.