Multiple minima in the free energy function

As pointed out by Bina (1998) the presence of multiple minima in the Gibbs free energy function may occur in a variety of cases and especially for systems that contain complex solid solutions and/or phases subject to abrupt phase transitions. Most constrained

optimization algorithms like FSQP do not deal with this case directly. A requirement for their use is that the objective function (or functions) is smoothly varying, and in practice this usually means that they will locate a minimum of Gsystem ‘in the neighborhood’ of the

starting guess. The FSQP algorithm adopted here cannot guarantee global minimal solutions, only local ones under the constraints imposed. In fact, no (practical) numerical method can guarantee a global minimum. Bina (1998) describes a simulated annealing algorithm for addressing the multiple minima problem. With FSQP this problem has been faced in the simplistic but, nevertheless, very effective way of setting up a routine to restart the algorithm from several different initial guesses. Solutions obtained are then compared. The attainment of identical solutions (i.e. same compositions and number of moles of phases at equilibrium) from different initial guesses makes it reasonable to assume that no local minima have been encountered and that the solution found represents the global minimum of Gsystem. The only negative aspect of this procedure is that using a ‘routine’ of

initial guesses can be quite time consuming from a computational point of view. For this reason ‘Gib’ has been left the option for the operator to use only one initial guess per time. In that case, before running the program, a file named ‘gib.initial’ has to be opened and values of site occupancies and number of moles for phases included in the computation have to be written. The file is then saved and it will be automatically called by ‘Gib’.

It is worth emphasizing here that in all applications of the program the ‘multiple minima problem’ has never appeared to be a real issue. The FSQP algorithm performs so effectively that regardless if the initial guess is a ‘good one’ (i.e. all the constraints satisfied and values of site occupancies and number of moles close to those of probable solution) or a very ‘bad one’ (i.e. with all the constraints violated and both site occupancies and number of moles set equal to zero), it generally converges to the correct global minimum routinely.

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