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.