Global minimization for problems with multiple local minima

We present a hybrid method for the minimization of the effective free energy of certain physical systems. The three stages of this method approximate the system interaction potential in three forms: (1) square-well, (2) quadratic, and (3) Lennard-Jones. The first stage uses a geometric method to minimize the free energy, based on a square-well potential. The second stage, assuming the pair interaction is quadratic, gives an analytical form for the minimization. The last stage, introducing more realistic physics, i.e., a Lennard-Jones pair interaction, uses the Monte Carlo method to perform minimization. The first stage is less accurate but much more efficient and eliminates most of the local minima of the free energy from further consideration. The refinement done by the latter steps reduces the error and delivers accurate results.