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Nonlinear Networks

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Financial Networks

Part of the book series: Advances in Spatial Science ((ADVSPATIAL))

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Abstract

Although Quesnay in 1758 conceptualized the financial flows in an economy as a network, the formal study of network flow problems dates to Kantorovich (1939), with the first applications being drawn from production and transportation/logistics problems (see also Hitchcock (1941) and Koopmans (1947)). Interestingly, such studies even preceded the development of linear programming with the work of Dantzig (1948) on the simplex method. The first network models were linear in that the costs on the links were assumed to be linear functions of the flows on the links.

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References

  • Ahlfeld, D. P., Dembo, R S., Mulvey, J. M., and Zenios, S. A., “Nonlinear Programming on Generalized Networks,” ACM Transactions on Mathematical Software 13 (1987) 350–368.

    Article  Google Scholar 

  • Ahuja, R. K., Magnanti, T. L., and Orlin, J. B., Network Flows Theory, Algorithms, and Applications, Prentice-Hall, Inc., Upper Saddle River, New Jersey, 1993.

    Google Scholar 

  • Aronson, J. E., “A Survey of Dynamic Network Flows,” Annals of Operations Research 20 (1989) 1–66.

    Article  Google Scholar 

  • Ball, M. O., Magnanti, T. L., Monma, C. L., and Nemhauser, G. L., editors, Network Models, Handbooks in Operations Research and Management Science, vol. 7, Elsevier Science B. V., Amsterdam, The Netherlands, 1995a.

    Google Scholar 

  • Ball, M. O., Magnanti, T. L., Monma, C. L., and Nemhauser, G. L., editors, Network Routing, Handbooks in Operations Research and Management Science, vol. 8, Elsevier Science B. V., Amsterdam, The Netherlands, 1995b.

    Google Scholar 

  • Beckmann, M. J., McGuire, C. B., and Winsten, C. B., Studies in the Economics of Transportation, Yale University Press, New Haven, Connecticut, 1956.

    Google Scholar 

  • Bertsekas, D. P., Constrained Optimization and Lagrange Multiplier Methods, Academic Press, Inc., New York, New York, 1982.

    Google Scholar 

  • Bertsekas, D., Castanon, D., Eckstein, J., and Zenios, S., “Parallel Computing in Network Optimization,” Network Models, Handbooks in Operations Research and Management Science, vol. 7, pp. 331–399, M. O. Ball, Magnanti, T. L., Monma, C. L., and Nemhauser, G. L., editors, Elsevier Science, B. V., Amsterdam. The Netherlands, 1995.

    Google Scholar 

  • Braess, D., “Uber ein Paradoxon der Verkehrsplanung,” Unternehmenforschung 12 (1968) 258–268.

    Google Scholar 

  • Dafermos, S., “Traffic Equilibrium and Variational Inequalities,” Transportation Science 14 (1980) 42–54.

    Article  Google Scholar 

  • Dafermos, S., “The General Multimodal Network Equilibrium Problem with Elastic Demand,” Networks 12 (1982) 57–72.

    Article  Google Scholar 

  • Dafermos, S., “An Iterative Scheme for Variational Inequalities,” Mathematical Programming 26 (1983) 40–47.

    Article  Google Scholar 

  • Dafermos, S., “Exchange Price Equilibria and Variational Inequalities,” Mathematical Programming 46 (1990) 391–402.

    Article  Google Scholar 

  • Dafermos, S., and Nagurney, A., “Isomorphism Between Spatial Price and Traffic Network Equilibrium Problems,” LCDS #85–17, Lefschetz Center for Dynamical Systems, Brown University, Providence, Rhode Island, 1985.

    Google Scholar 

  • Dafermos, S., and Nagurney, A., “Supply and Demand Equilibration Algorithms for a Class of Market Equilibrium Problems,” Transportation Science 23 (1989) 118–124.

    Article  Google Scholar 

  • Dafermos, S. C., and Sparrow, F. T., “The Traffic Assignment Problem for a General Network,” Journal of Research of the National Bureau of Standards 75B (1969) 91–117.

    Google Scholar 

  • Dantzig, G. B., “Programming in a Linear Structure,” in Comptroller, United States Air Force, Washington D.C., February 1948.

    Google Scholar 

  • Debreu, G., Theory of Value, Yale University Press, New Haven, Connecticut.

    Google Scholar 

  • Dembo, R. S., “A Primal Truncated Newton Algorithm for Large-Scale Nonlinear Network Optimization,” Mathematical Programming Study 31 (1987) 43–72.

    Article  Google Scholar 

  • Dembo, R. S., and Klincewicz, J. G., “Dealing with Degeneracy in Reduced Gradient Algorithms,” Mathematical Programming 31 (1985) 357–363.

    Article  Google Scholar 

  • Dembo, R. S., and Steihaug, T., “Truncated-Newton Algorithms for Large-Scale Unconstrained Optimization,” Mathematical Programming 26 (1983) 357–363.

    Article  Google Scholar 

  • Florian, M., and Hearn, D., “Network Equilibrium Models and Algorithms,” in Network Routing, Handbooks in Operations Research and Management Science, vol. 8, pp. 485–550, M. O. Ball, T. L. Magnanti, Monma, C. L., Nemhauser, G. L., editors, Elsevier Science B. V., Amsterdam, The Netherlands, 1995.

    Google Scholar 

  • Florian, M., and Los, M., “A New Look at Static Spatial Price Equilibrium Models,” Regional Science and Urban Economics 12 (1982) 579–597.

    Article  Google Scholar 

  • Gartner, N. H., “Optimal Traffic Assignment with Elastic Demands: A Review Part II: Algorithmic Approaches,” Transportation Science 14 (1980) 192–208.

    Article  Google Scholar 

  • Gill, P. E., Murray, W., and Wright, M. H., Practical Optimization, Academic Press, Inc., London, England, 1981.

    Google Scholar 

  • Hartman, P., and Stampacchia, G. “On some Nonlinear Elliptic Differential Functional Equations,” Acta Mathematica 115 (1966) 271–310.

    Article  Google Scholar 

  • Hitchcock, F. L., “The Distribution of a Product from Several Sources to Numerous Facilities,” Journal of Mathematical Physics 20 (1941) 224–230.

    Google Scholar 

  • Kantorovich, L. V., “Mathematical Methods in the Organization and Planning of Production,” Publication House of Leningrad University, Leningrad, USSR, 1939, translated in Management Science 6 (1960) 366–422.

    Article  Google Scholar 

  • Koopmans, T. C., “Optimum Utilization of the Transportation Systems,” Pro-ceedings of the International Statistical Conference, Washington, DC, 1947.

    Google Scholar 

  • Lasdon, L. S., Optimization Theory for Large Systems, The MacMillan Company, New York, New York, 1970.

    Google Scholar 

  • Markowitz, H. M., Portfolio Selection: Efficient Diversification of Investments, John Wiley & Sons, Inc., New York, New York, 1959.

    Google Scholar 

  • Mas-Colell, A., The Theory of General Economic Equilibrium: A Differentiable Approach, Econometric Society Monographs 9, Cambridge University Press, Cambridge, United Kingdom, 1985.

    Google Scholar 

  • Moore, C., and Nagurney, A., “A General Equilibrium Model of Interregional Monetary Flows,” Environment and Planning A (1989) 397–404.

    Article  Google Scholar 

  • Mulvey, J. M., “Nonlinear Networks in Finance,” Advances in Mathematical Programming and Financial Planning 1 (1987) 253–271.

    Google Scholar 

  • Murtagh, B., and Saunders, M., “Large-Scale Linearly Constrained Optimization,” Mathematical Programming 14 (1978) 41–72.

    Article  Google Scholar 

  • Nagurney, A., “An Equilibration Scheme for the Traffic Assignment Problem with Elastic Demand,” Transportation Research 22B (1988) 73–79.

    Google Scholar 

  • Nagurney, A., Network Economics: A Variational Inequality Approach, Kluwer Academic Publishers, Boston, Massachusetts, 1993.

    Book  Google Scholar 

  • Nagurney, A., and Eydeland, A., “A Splitting Equilibration Algorithm for the Computation of Large-Scale Constrained Matrix Problems: Theoretical Analysis and Application,” Computational Economics and Econometrics Advanced Studies in Theoretical and Applied Econometrics 22, pp. 65–105, H. M. Amman, D. Belsley, and L. Pau, editors, 1992.

    Chapter  Google 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.

    Article  Google Scholar 

  • Nagurney, A., and Robinson, A. G., “Algorithms for Quadratic Constrained Matrix Problems,” Mathematical and Computer Modelling 16 (1992) 53–65.

    Article  Google Scholar 

  • Nagurney A., and Zhang, D., Projected Dynamical Systems and Variational Inequalities with Applications, Kluwer Academic Publishers, Boston, Massachusetts, 1996a.

    Book  Google Scholar 

  • Nagurney, A., and Zhang, D., “Projected Dynamical Systems in the Formulation, Stability Analysis, and Computation of Fixed Demand Traffic Network Equilibria,” (1996b), to appear in Transportation Science.

    Google Scholar 

  • Quesnay, F., Tableau Economique, 1758, reproduced in facsimile with an introduction by H. Higgs by the British Economic Society, 1895.

    Google Scholar 

  • Samuelson, P. A., “Spatial Price Equilibrium and Linear Programming,” American Economic Review 42 (1952) 283–303.

    Google Scholar 

  • Sharpe, W. F., “A Simplified Model for Portfolio Analysis,” Management Science 9 (1963) 277–293.

    Article  Google Scholar 

  • Smith, M. J., “Existence, Uniqueness, and and Stability of Traffic Equilibria,” Transportation Research 13B (1979) 259–304.

    Google Scholar 

  • Soenen, L. A., Foreign Exchange Exposure Management: A Portfolio Approach, Sijthoff and Noordhoff, Germantown, Maryland, 1979.

    Google Scholar 

  • Takayama, T., and Judge, G. G., Spatial and Temporal Price and Allocation Models, North-Holland, Amsterdam, The Netherlands, 1971.

    Google Scholar 

  • Thore, S., Programming the Network of Financial Intermediation, Universitetsforlaget, Oslo, Norway, 1980.

    Google Scholar 

  • Thore, S., “Spatial Models of the Eurodollar Market,” Journal of Banking and Finance 8 (1984) 51–65.

    Article  Google Scholar 

  • Wald, A., “On Some Systems of Equations in Mathematical Economics,” Econometrica 19 (1951) 368–403.

    Article  Google Scholar 

  • Walras, L., Elements d’Economique Politique Pure, Corbaz, Lausanne, Switzerland, 1874.

    Google Scholar 

  • Wardrop, J. G., “Some Theoretical Aspects of Road Traffic Research,” in Proceedings of the Institute of Civil Engineers, Part II, pp. 25–378, 1952.

    Google Scholar 

Download references

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© 1997 Springer-Verlag Berlin Heidelberg

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Nagurney, A., Siokos, S. (1997). Nonlinear Networks. In: Financial Networks. Advances in Spatial Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-59066-5_5

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  • DOI: https://doi.org/10.1007/978-3-642-59066-5_5

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-63835-0

  • Online ISBN: 978-3-642-59066-5

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