Boltzmann Machines

Abstract

In this paper we have described the Boltzmann Machine and its potential use for problems from combinatorial optimization. Two remarks are worth mentioning here.

Firstly, the traveling salesman problem shows that it is not always easy to find a consensus function that is feasible and order preserving. In other words, the translation of the problem formulation into a provably equivalent Boltzmann Machine is generally nontrivial. In fact, for more complicated combinatorial optimization problems (e.g. job shop scheduling), one has not yet succeeded in designing a satisfactory Boltzmann Machine.

Secondly, the “cooling down scheme”, i.e. the way in which the control parameter c is decreased appears to be critical; to find a satisfactory cooling down scheme is a problem in itself.

In the past years however, a number of applications has appeared in literature, for which Boltzmann Machines were succesfully used. For an overview of these applications we refer to Aarts and Korst (1989). It is to be expected that Boltzmann Machines or similar approaches will play a major role in solving problems from combinatorial optimization. In fact, also due to the increasing availability of special purpose hardware, especially designed for Boltzmann Machines, its general importance will become more significant.