In this paper we consider a mathematical model for magmatic mixtures based on the Gibbs free energy. Different reformulations of the problem are presented and some theoretical results about the existence and number of solutions are derived. Finally, two homotopy methods and a global optimization one are introduced and computationally tested. One of the homotopy methods returns a single solution of the problem, while the other is able to return optimal solutions (often all of them). The global optimization method is a branch-and-reduce one with a theoretical guarantee of detecting all the solutions, although numerical difficulties may result in a loss of a few of them.