Application of the Nosé-Hoover method to optimization problems.

Abstract

A solution for continuous optimization problems is proposed using the Nosé-Hoover method. The proposed method aims for compatibleness, which has been a problem in many past solutions, between two requirements: searching with a high probability for finding candidates for the optimal points, and searching quickly in a feasible region. The Nosé-Hoover equation is used, where coordinates of a physical system are treated as the decision variables in a given optimization problem and a potential function is replaced by -k(B)T times the logarithm of an arbitrary density function for coordinate variables. The density can be set such that the visiting weight of the orbits to the equation has high values at areas where the objective function of the problem has low (high) values. Furthermore, a high value for the speed of the orbits can be set independently. Under an assumption of ergodicity, these values for the visiting weight and speed of the orbits are realized by long-time limits. Consequently, the two requirements can be satisfied. In numerical simulations assuming an objective function, the finite-time validity of the properties formulated with the long-time limits and the applicability of the proposed method to actual optimization problems were confirmed.