Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Some duality relationships for the generalized Neyman-Pearson problem


In 1963, Kuhn presented a dual problem to a relatively well-known location problem, variously referred to as the generalized Fermat problem and the Steiner-Weber problem. The purpose of this paper is to point out how Kuhn's results can be adapted to provide a dual to the generalized Neyman-Pearson problem, a problem of fundamental interest in statistics, which has applications in control theory and a number of other areas. The Neyman-Pearson problem, termed the dual problem, is a constrained maximization problem and may be considered to be a calculus-of-variations analog to the bounded-variable problem of linear programming. When the dual problem has equality constraints, the primal problem is an unconstrained minimization problem. Duality results are also obtained for the case where the dual problem has inequality constraints.

This is a preview of subscription content, log in to check access.


  1. 1.

    Kuhn, H.,Locational Problems and Mathematical Programming, Proceedings of the Colloquium on the Application of Mathematics to Economics, Budapest, 1963, Akademiai Kiado, Budapest, 1965.

  2. 2.

    Kuhn, H.,On a Pair of Dual Nonlinear Programs, Nonlinear Programming, Edited by J. Abadie, John Wiley and Sons, New York, 1967.

  3. 3.

    Neyman, J., andPearson, E.,Contributions to the Theory of Testing Statistical Hypotheses, Statistical Research Memoirs, Vol. 1, 1936.

  4. 4.

    Bellman, R., Glicksberg, I., andGross, O.,Some Aspects of the Mathematical Theory of Control Processes, The RAND Corporation, Report No. R-313, 1958.

  5. 5.

    Dantzig, G.,Linear Programming and Extensions, Princeton University Press, Princeton, New Jersey, 1963.

  6. 6.

    Karlin, S.,One-Stage Models with Uncertainty, Studies in the Mathematical Theory of Inventory and Production, K. J. Arrow, S. Karlin, and H. Scarf, Stanford University Press, Stanford, California, 1958.

  7. 7.

    Witzgall, C.,Optimal Location of a Central Facility: Mathematical Models and Concepts, National Bureau of Standards, Report No. 8388, 1965.

  8. 8.

    Bellman, R.,An Application of Dynamic Programming to Location-Allocation Problems, SIAM Review, Vol. 7, No. 1, 1965.

  9. 9.

    Francis, R.,Some Aspects of a Minimax Location Problem, Operations Research, Vol. 15, No. 6, 1967.

  10. 10.

    Kuhn, H., andKuenne, R.,An Efficient Algorithm for the Numerical Solution of the Generalized Weber Problem in Spatial Economics, Journal of Regional Science, Vol. 4, No. 1, 1962.

  11. 11.

    Kadane, J.,Discrete Search and the Neyman-Pearson Lemma, Journal of Mathematical Analysis and Applications, Vol. 22, No. 1, 1968.

  12. 12.

    Lehmann, E.,Testing Statistical Hypotheses, John Wiley and Sons, New York, 1959.

  13. 13.

    Dantzig, G., andWald, A.,On the Fundamental Lemma of Neyman and Pearson, Annals of Mathematical Statistics, Vol. 22. No. 1, 1951.

  14. 14.

    Chernoff, H., andScheffé, H.,A Generalization of the Neyman-Pearson Fundamental Lemma, Annals of Mathematical Statistics, Vol. 23, No. 2, 1952.

  15. 15.

    Virsan, C.,Sur le Lemme de Neyman-Pearson et la Programmation Linéaire, Comptes Rendus de l'Académie des Sciences de Paris, Series A-B, Vol. 263, No. 19, 1966.

  16. 16.

    Virsan, C.,The Neyman-Pearson Lemma and Linear Programming, Revue Roumaine de Mathématiques Pures et Appliquées, Vol. 12, No. 2, 1967.

  17. 17.

    Zahl, S.,An Allocation Problem with Applications to Operations Research and Statistics, Operations Research, Vol. 11, No. 3, 1963.

  18. 18.

    Capon, J.,On the Asymptotic Efficiency of Locally Optimum Detectors, IRE Transactions on Information Theory, Vol. 7, No. 2, 1961.

  19. 19.

    Francis, R.,Sufficient Conditions for Some Optimum-Property Facility Designs, Operations Research, Vol. 15, No. 3, 1967.

  20. 20.

    Reinhardt, H. E.,A Maximization Problem Suggested by Baker versus Carr, American Mathematical Monthly, Vol. 73, No. 10, 1966.

  21. 21.

    Wagner, D. H.,Nonlinear Functional Versions of the Neyman-Pearson Lemma, SIAM Review, Vol. 11, No. 1, 1969.

  22. 22.

    Beltrami, E. J.,A Penalty Approach to a Variational Lemma Concerning Optimum Allocation (to appear).

  23. 23.

    Eggleston, H. G.,Convexity, Cambridge University Press, Cambridge, England, 1958.

  24. 24.

    Rockafellar, R. T.,Duality in Nonlinear Programming, Mathematics of the Decision Sciences, Part 1, Edited by G. B. Dantzig and A. F. Veinott, Jr., American Mathematical Society, Providence, Rhode Island, 1968.

  25. 25.

    Rockafellar, R. T.,Duality and Stability in Extremum Problems Involving Convex Functions, Pacific Journal of Mathematics, Vol. 22, No. 1, 1967.

  26. 26.

    Karlin, S.,Mathematical Methods and Theory in Games, Programming and Economics, Vol. 2, Addison-Wesley Publishing Company, Reading, Massachusetts, 1959.

  27. 27.

    Royden, H. L.,Real Analysis, 2nd edition, The Macmillan Company, New York, 1968.

  28. 28.

    Kuhn, H., andTucker, A. W.,Nonlinear Programming, Proceedings of the 2nd Berkeley Symposium on Mathematical Statistics and Probability, University of California Press, Berkeley, California, 1951.

Additional Bibliography

  1. 29.

    Neyman, J., andPearson, E.,On the Problem of the Most Efficient Tests of Statistical Hypotheses, Philosophical Transactions of the Royal Society of London, Series A, Vol. 231, 1933.

  2. 30.

    Neyman, J., andPearson, E.,Joint Statistical Papers, University of California Press, Berkeley, California, 1967.

  3. 31.

    Nering, E. D.,Linear Algebra and Matrix Theory, John Wiley and Sons, New York, 1967.

  4. 32.

    Van-Slyke, R. M., andWets, R.,A Duality Theory for Abstract Mathematical Programs with Applications to Optimal Control Theory, Journal of Mathematical Analysis and Applications, Vol. 22, No. 3, 1968.

Download references

Author information

Additional information

This work was partially supported by the National Science Foundation, Grant Nos. NSF-GK-1571 and NSF-GK-3038. The authors would like to acknowledge the very useful comments of one of the referees, which led to more direct and general proofs of Properties 2.3 and 2.6.

Communicated by S. E. Dreyfus

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Francis, R.L., Wright, G.P. Some duality relationships for the generalized Neyman-Pearson problem. J Optim Theory Appl 4, 394–412 (1969).

Download citation


  • Control Theory
  • Minimization Problem
  • Equality Constraint
  • Fermat
  • Location Problem