In recent years there has been significant interest in adapting techniques from statistical physics, in particular mean field theory, to provide deterministic heuristic algorithms for obtaining approximate solutions to optimization problems. Although these algorithms have been shown experimentally to be successful there has been little theoretical analysis of them. In this paper we demonstrate connections between mean field theory methods and other approaches, in particular, barrier function and interior point methods. As an explicit example, we summarize 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 algorithms.