The softassign quadratic assignment algoithm is a discrete time, continuous state, synchronous updating optimizing neural network. While its effectiveness has been shown in the traveling salesman problem, graph matching and graph partitioning in thousands of simulations, there was no associated study of its convergence properties. Here, we construct discrete time Lyapunov functions for the cases of exact and approximate doubly stochastic constraint satisfaction which can be used to show convergence to a fixed point. The combination of good convergence properties and experimental success make the softassign algorithm the technique of choice for neural QAP optimization. 1 Introduction Discrete time optimizing neural networks are a well honed topic in neural computation. Beginning with the discrete state Hopfield model (Hopfield, 1982), considerable effort has been spent in analyzing the convergence properties of discrete time networks, especially along the dimensions of continuous versu...