The Gibbs tangent plane criterion has become important in determining the quality of obtained solutions to the phase and chemical equilibrium problem. The ability to determine if a postulated solution is thermodynamically stable with respect to perturbations in any or all of the phases is very useful in the search for the title equilibrium solution. Previous approaches have focused on finding stationary points of the tangent plane distance function. Obtaining all stationary points, however cannot be guaranteed due to the complex and nonlinear nature of the models used to predict equilibrium. Simpler fomulations for the stability problem are presented for special problems where nonideal liquid phases can be adequately modeled using the NRTL and UNIQUAC activity coefficient equations. It shows how the global minimum of the tangent plane distance function can be obtained for these problems. A global optimization approach is advantageous because a nonnegative solution can be asserted to be the globally stable equilibrium one, unlike available local algorithms. For the NRTL equation, the GOP algorithm of Floudas and Visweswaran (1990, 1993) is used to guarantee obtaining epsilon-global convergence to the global minimum. For the UNIQUAC equation, a branch and bound algorithm based on that of Falk and Soland (1969) is used to guarantee convergence to the global solution. Computational results demonstrate the efficiency of both global optimization algorithms in solving various challenging problems.