In recent years there has been significant interest in adapting techni
ques from statistical physics, in particular mean field theory, to pro
vide deterministic heuristic algorithms for obtaining approximate solu
tions to optimization problems. Although these algorithms have been sh
own experimentally to be successful there has been little theoretical
analysis of them. In this paper we demonstrate connections between mea
n field theory methods and other approaches, in particular, barrier fu
nction and interior point methods. As an explicit example, we summariz
e our work on the linear assignment problem. In this previous work we
defined a number of algorithms, including deterministic annealing, for
solving the assignment problem. We proved convergence, gave bounds on
the convergence times, and showed relations to other optimization alg
orithms.