Statistical mechanics approach in minimizing a multivariable function

Abstract

The method of minimization of a multivariable function based on the statistical mechanics analogy with a fictitious physical system of many particles is proposed. The function is assumed to be the Hamiltonian of the fictitious physical system to fit the global minimum of the function and the ground state ‘‘energy’’ of the fictitious system. In this model the global minimum search can be imitated by various relaxation processes in the fictitious system described by statistical mechanics. These relaxation processes lead to the equilibrium state, which is the ground state at the zero temperature limit. The imitation of a relaxation process confers to the minimization procedure the advantage of a relaxation process in a real physical system: because of thermal fluctuations a real system cannot be trapped by metastable states related to local minima. It always reaches the equilibrium state. The simulations of the relaxation processes based on the macroscopic kinetic equations and on the Monte Carlo algorithms are discussed. The new Monte Carlo algorithm based on the simulation of random walks of the representative point of the system in multidimensional phase space of the variables of the function under investigation is proposed. Unlike the conventional Metropolis–Rosenbluths–Tellers Monte Carlo method, each elementary transition in the proposed algorithm results in simultaneous movement of all atoms of the system, i.e., it generates a fluctuation involving any number of atoms.