Constrained Matrix Problems

  • Anna Nagurney
Part of the Advances in Computational Economics book series (AICE, volume 10)


In this chapter the focus is on constrained matrix problems, the solutions of which are needed to create base-line datasets for use in equilibrium modeling. The constrained matrix problem is to compute the best possible estimate of an unknown matrix, given some information to constrain the solution set, and requiring that the matrix be a minimum distance from a given matrix. The problem arises as a core problem in numerous applications, including: the estimation of input/output tables, social/national accounts, and financial flow of funds accounts, the projection of migration flows over space and time, the projection of origin/destination transportation flows, and the estimation of contingency tables in statistics.


Column Total Financial Flow Matrix Problem Column Equilibration Data Level Parallelism 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Bacharach, M., Biproportional Scaling and Input-Output Change, Cambridge University Press, Cambridge, United Kingdom, 1970.Google Scholar
  2. Bachem, A., and Korte, B., “Algorithm for quadratic optimization over transportation polytopes,” Zeitschrift fur Angewandte Mathematik und Mechanik 58 (1978) T459–T461.CrossRefGoogle Scholar
  3. Bregman, L. M., “Proof of the convergence of Sheleikhovskii’s method for a problem with transportation constraints,” USSR Computational Mathematics and Mathematical Physics 1 (1967) 191–204.CrossRefGoogle Scholar
  4. Byron, R. P., “The estimation of large social accounts matrices,” Journal of the Royal Statistical Society Series A 141 (1978) 359–369.CrossRefGoogle Scholar
  5. Cottle, R. W., Duvall, S. G., and Zikan, K., “A lagrangean relaxation algorithm for the constrained matrix problem,” Naval Research Logistics Quarterly 33 (1986) H-76.Google Scholar
  6. Deming, W. E., and Stephan, F. F., “On a least-squares adjustment of a sampled frequency table when the expected marginal totals are known,” Annals of Mathematical Statistics 11 (1940) 427–444.CrossRefGoogle Scholar
  7. Eydeland, A., and Nagurney, A., “Progressive equilibration algorithms: The case of linear trac saction costs,” Computer Science in Economics and Management 2 (1989) 197–219.CrossRefGoogle Scholar
  8. Florian, M., “Nonlinear cost network models in transportation analysis,” Mathematical Program ping Study 26 (1986) 167–196.CrossRefGoogle Scholar
  9. Friedlander, D., “A technique for estimating a contingency table given the marginal total and some supplementary data,” Journal of the Royal Statistical Society A 124 (1961) 412–420.CrossRefGoogle Scholar
  10. Hanson, K. A., and Robinson, S, S., “Data, linkages, and models: U.S. national income and product accounts in the framework of a social accounting matrix,” Agriculture and Rural Economy Division, Economic Research Service, U. S. Department of Agriculture, Staff Report No. AGES 89-5, 1989.Google Scholar
  11. Harrigan, F., and Buchanan, I., “Quadratic programming approach to input-output estimation and simulation,” Journal of Regional Science 24 (1984) 339–358.CrossRefGoogle Scholar
  12. Hughes, M., and Nagurney, A., “A network model and algorithm for the analysis and estimation of financial flow of funds,” Computer Science in Economics and Management 5 (1992) 23–39.CrossRefGoogle Scholar
  13. Judge, G. G., and Yancey, T. A., Improved Methods of Inference in Econometrics, North-Holland, Amsterdam, The Netherlands, 1986.Google Scholar
  14. Kim, D. S., and Nagurney, A., “Massively parallel implementation of the Splitting Equilibration Algorithm,” Computer Science in Economics and Management 6 (1993) 151–161.Google Scholar
  15. King, B. B., “What is a SAM?” in Social Accounting Matrices: A Basis for Planning, G. Pyatt and J. I. Round, editors, The World Bank, Washington, DC, 1985.Google Scholar
  16. Klincewicz, J., “Implementation of an exact Newton method for separable convex transportation problems,” Networks 19 (1989) 95–105.CrossRefGoogle Scholar
  17. Kruithof, J., “Telefoonverkeersrekening,” De Ingenieur 3 (1937) 15–25.Google Scholar
  18. Nagurney, A., “An algorithm for the solution of a quadratic program- ming problem with application to constrained matrix and spatial price equilibrium problems,” Environment and Planning A 21 (1989) 99–114.CrossRefGoogle Scholar
  19. Nagurney, A., and Eydeland, A., “A Splitting Equilibration Algorithm for the computation of large-scale constrained matrix problems: theoretical analysis and application,” in Computational Economics and Econometrics, Advanced Studies in Theoretical and Applied Econometrics 22, pp. 65–105, H. M. Amman, D. A. Belsley, and L. F. Pau, editors, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1992a.Google Scholar
  20. Nagurney, A., and Eydeland, A., “A parallel network equilibration algorithm for a class of constrained matrix problems,” Transportation Science 26 (1992b) 59–68.CrossRefGoogle Scholar
  21. Nagurney, A., and Hughes, M., “Financial flow of funds networks,” Networks 22 (1992) 145–161.CrossRefGoogle Scholar
  22. Nagurney, A., Kim, D. S., and Robinson, A. G., “Serial and parallel equilibration of large-scale constrained matrix problems with application to the social and economic sciences,” The International Journal of Supercomputer Applications 4. 1 (1990) 49–71.Google Scholar
  23. Nagurney, A., and Robinson, A. G., “Algorithms for quadratic constrained matrix problems,” Mathematical and Computer Modelling 16 (1992) 53–65.CrossRefGoogle Scholar
  24. Ohuchi, A., and Kaji, I., “Lagrangean dual coordinatewise maximization for network transportation problems with quadratic costs,” Networks 14 (1984) 525–530.CrossRefGoogle Scholar
  25. Polenske, K. E., U. S. Multiregional Input-Output Accounts and Model, Lexington Books, Lexington, Massachusetts, 1980.Google Scholar
  26. Quesnay, F., Tableau Economique, 1758, reproduced in facsimile with an introduction by H. Higgs by the British Economic Society, 1895.Google Scholar
  27. Schneider, M. H., and Zenios, S., “A comparative study of algorithms for matrix balancing,” Operations Research 38 (1990) 439–455.CrossRefGoogle Scholar
  28. Stone, R., “Simple transaction models, information, and computing,” The Review of Economic Studies XIX, 49 (1951) 67–84.CrossRefGoogle Scholar
  29. Stone, R., “Multiple classifications if social accounting,” Bulletin de l’Institut International de Statistique 39 (1962) 215–233.Google Scholar
  30. Van der Ploeg, F., “Reliability and adjustment of sequences of large economic accounting matrices,” Journal of the Royal Statistical Society A 145 (1982) 169–184.Google Scholar
  31. Van der Ploeg, F., “Balancing large systems of national accounts,” Computer Science in Economics and Management 1 (1988) 31–39.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1999

Authors and Affiliations

  • Anna Nagurney
    • 1
  1. 1.University of MassachusettsAmherstUSA

Personalised recommendations