This paper reports both theoretical results and also computational experience with a method for approximating a competitive equilibrium in a piecewise linear economy. The algorithm consists of solving a sequence of linear programs, alternating between: (a) a "master" problem which ensures a balancing bundle of choices and generates a price vector; and (b) a "sub" problem which indicates the maximum level of utility attainable by each household--given the initial resource endowments--and also given the prices generated at the current iteration of the master problem. Each subproblem provides a "price-consistent" utility vector. The master problem determines a convex combination of the utility vectors generated at previous iterations. This convex combination mimimize the distance between the quantity-consistent and the price-consistent set. For the sequence of sub and master problems to approach a competitive equilibrium, this distance must approach zero. Thus far, the algorithm has failed whenever all equilibria are "unstable." and it has converged rapidly when there are "suitable" equilibria. It will be shown that the algorithm does not cycle. It will also be shown that if the sequence of solutions (obtained from the algorithm) converges, then it converges to a Walrasian equilibrium.