Network Economics pp 347-391 | Cite as

# Constrained Matrix Problems

## Abstract

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.

## Keywords

Column Total Financial Flow Matrix Problem Column Equilibration Data Level Parallelism## Preview

Unable to display preview. Download preview PDF.

## References

- Bacharach, M.,
**Biproportional Scaling and Input-Output Change**, Cambridge University Press, Cambridge, United Kingdom, 1970.Google Scholar - Bachem, A., and Korte, B., “Algorithm for quadratic optimization over transportation polytopes,”
*Zeitschrift fur Angewandte Mathematik und Mechanik***58**(1978) T459–T461.CrossRefGoogle Scholar - 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 - Byron, R. P., “The estimation of large social accounts matrices,”
*Journal of the Royal Statistical Society Series A***141**(1978) 359–369.CrossRefGoogle Scholar - 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 - 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 - 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 - Florian, M., “Nonlinear cost network models in transportation analysis,”
*Mathematical Program ping Study***26**(1986) 167–196.CrossRefGoogle Scholar - 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 - 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 - Harrigan, F., and Buchanan, I., “Quadratic programming approach to input-output estimation and simulation,”
*Journal of Regional Science***24**(1984) 339–358.CrossRefGoogle Scholar - 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 - Judge, G. G., and Yancey, T. A.,
**Improved Methods of Inference in Econometrics**, North-Holland, Amsterdam, The Netherlands, 1986.Google Scholar - 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 - 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 - Klincewicz, J., “Implementation of an exact Newton method for separable convex transportation problems,”
*Networks***19**(1989) 95–105.CrossRefGoogle Scholar - 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 - 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 - 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 - Nagurney, A., and Hughes, M., “Financial flow of funds networks,”
*Networks***22**(1992) 145–161.CrossRefGoogle Scholar - 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 - Nagurney, A., and Robinson, A. G., “Algorithms for quadratic constrained matrix problems,”
*Mathematical and Computer Modelling***16**(1992) 53–65.CrossRefGoogle Scholar - Ohuchi, A., and Kaji, I., “Lagrangean dual coordinatewise maximization for network transportation problems with quadratic costs,”
*Networks***14**(1984) 525–530.CrossRefGoogle Scholar - Polenske, K. E., U.
**S. Multiregional Input-Output Accounts and Model**, Lexington Books, Lexington, Massachusetts, 1980.Google Scholar - Quesnay, F.,
**Tableau Economique**, 1758, reproduced in facsimile with an introduction by H. Higgs by the British Economic Society, 1895.Google Scholar - Schneider, M. H., and Zenios, S., “A comparative study of algorithms for matrix balancing,”
*Operations Research***38**(1990) 439–455.CrossRefGoogle Scholar - Stone, R., “Simple transaction models, information, and computing,”
*The Review of Economic Studies XIX*, 49 (1951) 67–84.CrossRefGoogle Scholar - Stone, R., “Multiple classifications if social accounting,”
*Bulletin de l’Institut International de Statistique***39**(1962) 215–233.Google Scholar - 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 - Van der Ploeg, F., “Balancing large systems of national accounts,”
*Computer Science in Economics and Management***1**(1988) 31–39.Google Scholar