Binary optimization: On the probability of a local minimum detection in random search

Abstract

The problem of binary optimization of a quadratic functional is discussed. By analyzing the generalized Hopfield model we obtain expressions describing the relationship between the depth of a local minimum and the size of the basin of attraction. Based on this, we present the probability of finding a local minimum as a function of the depth of the minimum. Such a relation can be used in optimization applications: it allows one, basing on a series of already found minima, to estimate the probability of finding a deeper minimum, and to decide in favor of or against further running the program. The iterative algorithm that allows us to represent any symmetric N ×N matrix as a weighted Hebbian series of bipolar vectors with a given accuracy is proposed. It so proves that all conclusions about neural networks and optimization algorithms that are based on Hebbian matrices are true for any other type of matrix. The theory is in a good agreement with experimental results. © 2008 Springer-Verlag Berlin Heidelberg.