Locating global minima in optimization problems by a random-cost approach

Abstract

OPTIMIZATION problems are common to many diverse disciplines. The classic example is the travelling-salesman problem^1,2, in which the objective is to find the shortest path connecting a number of cities. Spin glasses³, meanwhile, exemplify a general class of systems subject to conflicting constraints: in this case, the impossibility of each spin in a lattice aligning favourably with all of its neighbours leads to 'frustration' and to a large number of local energy minima. The optimization problem then becomes the attempt to find the global minimum, or ground state. Predicting the conformational ground state of proteins⁴ is closely allied to the spin-glass problem. The chief difficulty in searching for global minima is that straightforward search algorithms¹ tend to become trapped in local minima. Only a few promising approaches, such as simulated annealing² or genetic algorithms⁵, exist. Here I present a general purpose Monte Carlo procedure in which local redirections of the search path are effected at all relevant length scales while enforcing a one-dimensional random walk in the function being minimized. This method yields a series of extrema, from which the global minimum can be extracted with high probability in the limit of large statistics.