Computer Science in Economics and Management

, Volume 2, Issue 3, pp 197–219 | Cite as

Progressive equilibration algorithms: The case of linear transaction costs

  • Alexander Eydeland
  • Anna Nagurney


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.

Key words

Market equilibrium spatial price algorithm progressive equilibration 


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  1. Asmuth, R., Eaves, B.C., and Peterson, E.L. (1979), Computing economic equilibria on affine networks with Lemke's algorithm, Mathematics of Operations Research 4, 209–214.Google Scholar
  2. Dafermos, S. (1986), Isomorphic multiclass spatial price and multimodal traffic network equilibrium models, Regional Science and Urban Economics 16, 197–209.Google Scholar
  3. Defermos, S. and Nagurney, A. (1987), Supply and demand equilibration algorithms for a class of market equilibrium problems, forthcoming in Transportation Science.Google Scholar
  4. Florian, M. and Los, M. (1982), A new look at static spatial price equilibrium models, Regional Science and Urban Economics 12, 579–597.Google Scholar
  5. Glassey, C.R. (1978), A quadratic network optimization model for equilibrium single commodity trade flows, Mathematical Programming 14, 98–107.Google Scholar
  6. Güder, F. (1987), Pairwise reactive SOR algorithm for quadratic programming of net import spatial equilibrium models, forthcoming in Mathematical Programming.Google Scholar
  7. Jones, P.C., Saigal, R., and Schneider, M.H. (1986), A variable dimension homotopy on networks for computing linear spatial equilibria, Discrete Applied Mathematics 13, 131–156.Google Scholar
  8. Judge, G.G., and T. Takayama (1973), Studies in Economic Planning over Space and Time, North Holland Publishing Co.Google Scholar
  9. Nagurney, A. (1987a), An algorithm for the classical spatial price equilibrium problem, Operations Research Letters 6(2), 93–98.Google Scholar
  10. Nagurney, A. (1987b), Computational comparisons of spatial price equilibrium methods, Journal of Regional Science 27, 55–76.Google Scholar
  11. Nagurney, A. (1987c), Competitive equilibrium problems, variational inequalities, and regional science, Journal of Regional Science 27, 503–514.Google Scholar
  12. Nagurney, A. (1988a) An algorithm for the solution of a quadratic programming problem with application to constrained matrix and spatial price equilibrium problems, Environment & Planning A in press.Google Scholar
  13. Nagurney, A. (1988b), Import and export equilibration algorithms for the solution of the net import model, Journal of Cost Analysis in press.Google Scholar
  14. Nagurney, A. and Aronson, J. (1988), A general dynamic spatial price equilibrium model: formulation, solution, and computational results, Journal of Computational and Applied Mathematics 359–377.Google Scholar
  15. Nagurney, A. and Kim, D.S. (1988), Parallel and serial variational inequality decomposition algorithms for multicommodity market equilibrium problems, The International Journal of Supercomputer Applications in press.Google Scholar
  16. Press, W.H., Flannery, B.H., Teukolsky, S.A., and Vetterling, W.T. (1986), Numerical Recipes, Cambridge University Press.Google Scholar
  17. Samuelson, P.A. (1952), Spatial price equilibrium and linear programming, American Economic Review 42, 283–303.Google Scholar
  18. Takayama, T. and Judge, G.G. (1971), Spatial and Temporal Price and Allocation Models, North-Holland, Amsterdam.Google Scholar
  19. Zangwill, W.I. (1969), Nonlinear Programming — A Unified Approach, Prentice Hall.Google Scholar

Copyright information

© Kluwer Academic Publishers 1989

Authors and Affiliations

  • Alexander Eydeland
    • 1
  • Anna Nagurney
    • 2
  1. 1.Department of Mathematics and StatisticsUniversity of MassachusettsAmherstUSA
  2. 2.Department of General Business & Finance, School of ManagementUniversity of MassachusettsAmherstUSA

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