Optimization and homotopy methods for the Gibbs free energy of simple magmatic mixtures

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 multiple 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 some numerical difficulties, resulting in a loss of a few of them, may have to be faced.