Optimization Letters

, Volume 9, Issue 6, pp 1053–1061 | Cite as

Ragnar Frisch and interior-point methods

  • Olav Bjerkholt
  • Sjur Didrik Flåm
Original Paper


The distinguished econometrician Ragnar Frisch (1895–1973) also played an important role in optimization theory. In fact, he was a pioneer of interior-point methods. This note reconsiders his contribution, relating it to history and modern developments.


Linear program Logarithmic potential Interior-point methods 


  1. 1.
    Arrow, K.J.: The work of Ragnar Frisch, econometrician. Econometrica 28(2), 175–192 (1960)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bjerkholt, O.: Some unresolved problems of mathematical programming. In: Basu, D. (ed.) Economic models–methods, theory and applications, pp. 3–19. World Scientific, Singapore (2009)Google Scholar
  3. 3.
    Carroll, C.W.: The created response surface technique for optimizing nonlinear restrained systems. Oper. Res. 9(2), 169–185 (1961)CrossRefzbMATHGoogle Scholar
  4. 4.
    Dantzig, G.B.: Programming of interdependent activitities: II mathematical model. Econometrica 17, 200–211 (1949)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Dantzig, G.B.: Linear programming and extensions. Princeton University Press, Princeton (1963)zbMATHGoogle Scholar
  6. 6.
    Dantzig, G.B.: The diet problem. Interfaces 20(4), 43–47 (1990)CrossRefGoogle Scholar
  7. 7.
    Dikin, I.: Iterative solution of problems of linear and quadratic programming. Sov. Math. Dokl. 8, 674–675 (1967)zbMATHGoogle Scholar
  8. 8.
    Dorfman, R., Samuelson, P.A., Solow, R.M.: Linear programming and economic analysis. McGraw-Hill, New York (1958)zbMATHGoogle Scholar
  9. 9.
    Ferris, M., Mangasarian, O. L., Wright, S. J.: Linear programming with MATLAB, MPS-SIAM series on optimization (2007)Google Scholar
  10. 10.
    Fiacco, A.V., McCormick, G.P.: Computational algorithm for the sequential unconstrained minimization technique for nonlinear programming. Manag. Sci. 10(4), 601–617 (1964)CrossRefGoogle Scholar
  11. 11.
    Fiacco, A.V., McCormick, G.P.: Nonlinear programming, sequential unconstrained minimization techniques, Research Analysis Corporation (1968) and SIAM (1990)Google Scholar
  12. 12.
    Frisch, R.: Problems and methods of econometrics. In: Bjerkholt, O., Dupont-Kieffer, A. (eds.) The Poincaré lectures 1933. Routledge, London (2009)Google Scholar
  13. 13.
    Frisch, R.: Circulation planning: proposal for a national organization of a commodity and service exchange. Econometrica 2, 258–336 and 422–435 (1934)Google Scholar
  14. 14.
    Frisch, R.: Introduction. In: Wold, K. (ed.) Kosthold og levestandard, en økonomisk undersøkelse (Nutrition and standard of living, an economic investigation), pp. 1–13. Fabritius og Sønners Forlag, Oslo (1941)Google Scholar
  15. 15.
    Frisch, R.: Das Ausleseproblem in der Bienenzüchtung. Zeitschrift für Bienenforschung 1, 7 (1952)Google Scholar
  16. 16.
    Frisch, R.: Principles of linear programming—with particular reference to the double gradient form of the logarithmic potential method. Memorandum from the Institute of Economics, University of Oslo, Oslo (1954)Google Scholar
  17. 17.
    Frisch, R.: The logarithmic potential method of convex programming with particular application to the dynamics of planning for national development. Memorandum from the Institute of Economics, University of Oslo, Oslo (1955)Google Scholar
  18. 18.
    Frisch, R.: La résolution des problèmes de programmes linéaires par la méthode du potential logarithmique, Cahiers du Séminaire D’Économetrie, No 4—Programme linéaire—Agrégation et nombre indices 7–23 (1956a)Google Scholar
  19. 19.
    Frisch, R.: Formulazione di un piano di sviluppo nazional come problema di programmazione convessa, L’industria (1956b)Google Scholar
  20. 20.
    Frisch, R.: Macroeconomics and linear programming. In: 25 Economic Essays in Honour of Erik Lindahl, Ekonomisk Tidskrift, pp. 38–67 Stockholm (1956c).Google Scholar
  21. 21.
    Frisch, R.: The multiplex method for linear programming. Sankhya 18(3&4), 329–362 (1957)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Gale, D.: The theory of linear economic models. The University of Chicago Press, Chicago (1960)Google Scholar
  23. 23.
    Gondzio, J., Terlaky, T.: A computational view of interior-point methods for linear programming. In: Beasley, J. (ed.) Advances in linear and integer programming, Ch. 3, pp. 103–144. Oxford University Press, Oxford (1996)Google Scholar
  24. 24.
    Hitchcock, F.L.: The distribution of a product from several sources to numerous localities. J. Math. Phys. 20, 224–230 (1941)MathSciNetGoogle Scholar
  25. 25.
    Huard, P.: Resolution of mathematical programming with nonlinear constraints by the method of centers. North-Holland, Amsterdam (1967)Google Scholar
  26. 26.
    Kantorovich, L.V.: A new method of solving some classes of extremal problems. Dokl. Akad Sci USSR 28, 211–214 (1940)Google Scholar
  27. 27.
    Karmarkar, N.: A new polynomial-time algorithm for linear programming. Combinatorica 4(4), 373–395 (1984)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Khachiyan, L.G.: A polynomial algorithm for linear programming. Sov. Math. Dokl. 20, 191–194 (1979)zbMATHGoogle Scholar
  29. 29.
    Klee, V., Minty, G.J.: How good is the simplex algorithm, in inequalities III, pp. 159–172. Academic Press, New York (1972)Google Scholar
  30. 30.
    Koopmans, T.C. (ed.): Activity analysis of production and allocation, monograph 13 Cowles commision for research in economics. Wiley, Hoboken (1951)Google Scholar
  31. 31.
    Leontief, W.W.: Input-output economics, vol. 2. Oxford University Press, New York (1986)Google Scholar
  32. 32.
    Lootsma, F.A.: Hessian matrices of penalty functions for solving constrained optimization problems. Philips Res. Rep. 24, 322–331 (1969)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Luenberger, D.: Introduction to linear and nonlinear programming. Addison-Wesley, Boston (1984)Google Scholar
  34. 34.
    Marsten, R., Subramanian, R., Saltzman, M., Lustig, I., Shanno, D.: Interior point methods for linear programming: just call Newton, Lagrange, and Fiacco and McCormick. Interfaces 20(4), 105–116 (1990)Google Scholar
  35. 35.
    Meggido, N.: Pathways to the optimal set in linear programming. In: Meggido, N. (ed.) Progress in mathematical programming: interior-point and related methods, Chap 8, pp. 131–158. Springer, Berlin (1989)CrossRefGoogle Scholar
  36. 36.
    Murray, W.: Analytical expressions for the eigenvalues and eigenvectors of the Hessian matrices of barrier and penalty functions. J. Optim. Theory Appl. 7, 189–196 (1971)CrossRefzbMATHGoogle Scholar
  37. 37.
    Nash, S.G.: SUMT (revisited). Oper. Res. 46, 763–775 (1998)CrossRefzbMATHGoogle Scholar
  38. 38.
    Nocedal, J., Wright, S.J.: Numerical optimization. Springer, Berlin (2006)Google Scholar
  39. 39.
    Prékopa, A.: Contributions to the theory of stochastic programming. Math. Program. 4, 202–221 (1973)CrossRefzbMATHGoogle Scholar
  40. 40.
    Sandmo, A.: Ragnar Frisch on the optimal diet. Hist. Polit. Econ. 25, 313–327 (1993)CrossRefGoogle Scholar
  41. 41.
    Schrijver, A.: Theory of linear and integer programming. Wiley, New York (1986)zbMATHGoogle Scholar
  42. 42.
    Shanno, D.: Who invented the interior-point method? Documenta mathematica, optimization stories, 21st ISMP Berlin, pp. 55–64 (2012)Google Scholar
  43. 43.
    Sonnevend, G.: An “analytic center” for polyhedrons and new classes of global algorithms for linear (smooth, convex) programming. In: Prekopa, A., Szelezsan, J., Strazicky, B.: (eds.) System modelling and optimization, Proceedings 12th IFIP Conference Lecture Notes in Control and Information Sciences 84, pp. 866–876 Springer, Berlin (1985)Google Scholar
  44. 44.
    Sporre, G.: On some properties of interior point methods for optimization, Doctoral Thesis, Royal Institute of Technology, Stockholm (2003)Google Scholar
  45. 45.
    Stiefel, E.: Note on Jordan elimination, linear programming and Tchebycheff approximation. Numerische Mathematik 2, 1–17 (1960)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Stiegler, G.J.: The cost of subsistence. J. Farm Econ. 27(2), 303–314 (1941)CrossRefGoogle Scholar
  47. 47.
    Vanderbei, R.J.: Linear programming. Kluwer, Boston (1996)zbMATHGoogle Scholar
  48. 48.
    van Eijk, C.J., Sandee, J.: Quantitative determination of an optimum economic policy. Econometrica 27, 1–13 (1959)CrossRefzbMATHGoogle Scholar
  49. 49.
    Wright, M.H.: Ill-conditioning and computational error in interior methods for nonlinear programming. SIAM J. Optim. 9, 84–111 (1999)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Economics DepartmentUniversity of OsloOsloNorway
  2. 2.Economics DepartmentUniversity of BergenBergenNorway

Personalised recommendations