Progressive equilibration algorithms: The case of linear transaction costs

Abstract

In this paper we consider the solution of large-scale market equilibrium problems with linear transaction costs which can be formulated as strictly convex quadratic programming problems, subject to supply and demand constraints. In particular, we introduce two new classes of progressive equilibration algorithms, which retain the simplicity of the original cyclic ones in that at each step either the supply or demand market equilibrium subproblem can be solved explicitly in closed form. However, rather than equilibrating the markets in cyclic manner, the next market to be equilibrated is selected in a more strategic fashion. We then provide qualitative results for the entire family of progressive equilibration algorithms, i.e., the rate of convergence and computational complexity. We discuss implementation issues and give computational results for large-scale examples in order to illustrate and give insights into the theoretical analysis. Furthermore, we show that one of the new classes of algorithms, the ‘good-enough’ one, is computationally the most efficient. Theoretical results are important in that the relative efficiency of different algorithms need no longer be language, machine, or programmer dependent. Instead, the theory can guide both practitioners and researchers in ensuring that their implementation of these algorithms is, indeed, good. Since an equivalent quadratic programming problem arises in a certain class of constrained matrix problems, our results can be applied there, as well. Finally, since more general asymmetric multicommodity market equilibrium problems can be solved as series of the type of problems considered here, the result$ are also applicable to such equilibrium problems.