Optimization Letters

, Volume 4, Issue 2, pp 275–285 | Cite as

Interpretation of Lagrange multipliers in nonlinear pricing problem

  • Kimmo Berg
  • Harri Ehtamo
Original Paper


We present well-known interpretations of Lagrange multipliers in physical and economic applications, and introduce a new interpretation in nonlinear pricing problem. The multipliers can be interpreted as a network of directed flows between the buyer types. The structure of the digraph and the fact that the multipliers usually have distinctive values can be used in solving the optimization problem more efficiently. We also find that the multipliers satisfy a conservation law for each node in the digraph, and the non-uniqueness of the multipliers are connected to the stability of the solution structure.


Lagrange multipliers Nonlinear pricing Flow network Conservation law Stability Sensitivity 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Basov S.: Multidimensional Screening. Springer, Heidelberg (2005)zbMATHGoogle Scholar
  2. 2.
    Berg, K., Ehtamo, H.: Multidimensional screening: online computation and limited information. In: ICEC 2008: Proceedings of the 10th International Conference on Electronic Commerce. ACM Int. Conf. Proc. Ser. 42, Innsbruck, Austria (2008).
  3. 3.
    Berg K., Ehtamo H.: Learning in nonlinear pricing with unknown utility functions. Ann. Oper. Res. 172(1), 375–392 (2009)zbMATHCrossRefGoogle Scholar
  4. 4.
    Bertsekas D.P.: Nonlinear Programming. Athena Scientific, Belmont (2004)Google Scholar
  5. 5.
    Bertsekas D.P., Nedic A., Ozdaglar A.E.: Convex Analysis and Optimization. Athena scientific, Belmont (2003)zbMATHGoogle Scholar
  6. 6.
    Boyd S., Vandenberghe L.: Convex Optimization. Cambridge University Press, New York (2004)zbMATHGoogle Scholar
  7. 7.
    Bussotti P.: On the genesis of the Lagrange multipliers. J. Optim. Theory Appl. 117, 453–459 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Ehtamo, H., Berg, K., Kitti, M.: An adjustment scheme for nonlinear pricing problem with two buyers. Eur. J. Oper. Res. (2009). doi: 10.1016/j.ejor.2009.01.037.
  9. 9.
    Fiacco A.V., Liu J.: Degeneracy in NLP and the development of results motivated by its presence. Ann. Oper. Res. 46, 61–80 (1993)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Garille S.G., Gass S.I.: Stigler’s diet problem revisited. Oper. Res. 49(1), 1–13 (2001)CrossRefGoogle Scholar
  11. 11.
    Nahata, B., Kokovin, S., Zhelobodko, E.: Self-selection under non-ordered valuations: type-splitting, envy-cycles, rationing and efficiency. Working paper (2001)Google Scholar
  12. 12.
    Rochet J.-C., Stole L.A.: The economics of multidimensional screening. In: Dewatripont, M., Hansen, L.P., Turnovsky, S.J. (eds) Advances in Economics and Econometrics, vol. 1, pp. 150–197. Cambridge university press, Cambridge (2003)CrossRefGoogle Scholar
  13. 13.
    Stigler G.J.: The cost of subsistence. J. Farm Econ. 27(2), 303–314 (1945)CrossRefGoogle Scholar
  14. 14.
    Wilson R.B.: Nonlinear Pricing. Oxford University Press, Oxford (1993)Google Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Systems Analysis LaboratoryHelsinki University of TechnologyTKKFinland

Personalised recommendations