Merging Constrained Optimisation with Deterministic Annealing to "Solve" Combinatorially Hard Problems

Several parallel analogue algorithms, based upon mean field theory (MFT) approximations to an underlying statistical mechanics formulation, and requiring anexternally prescribed annealing schedule, now exist for finding approximate solutions to difficult combinatorial optimisation problems. They have been applied to the Travelling Salesman Problem (TSP), as well as to various issues in computational vision and cluster analysis. I show here that any given MFT algorithm can be combined in a natural way with notions from the areas of constrained optimisation and adaptive simulated annealing to yield a single homogenous and efficient parallel relaxation technique,for which an externally prescribed annealing schedule is no longer required. The results of numerical simulations on 50-city and 100-city TSP problems are presented, which show that the ensuing algorithms aretypically an order of magnitude faster than the MFT algorithms alone, and which also show, on occasion, superior solutions as well. 1 INTRODUCTION Several promising parallel analogue algorithms, which can be loosely described by the term "deterministic annealing", or "mean field theory (MFT) annealing", have *also at Theoretical Division and Center for Nonlinear Studies, MSB213, Los Alamos National Laboratory, Los Alamos, NM 87545.