Journal of Optimization Theory and Applications

, Volume 4, Issue 6, pp 394–412 | Cite as

Some duality relationships for the generalized Neyman-Pearson problem

  • Richard L. Francis
  • Gordon P. Wright
Contributed Papers


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.


Control Theory Minimization Problem Equality Constraint Fermat Location Problem 
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  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.Google Scholar
  2. 2.
    Kuhn, H.,On a Pair of Dual Nonlinear Programs, Nonlinear Programming, Edited by J. Abadie, John Wiley and Sons, New York, 1967.Google Scholar
  3. 3.
    Neyman, J., andPearson, E.,Contributions to the Theory of Testing Statistical Hypotheses, Statistical Research Memoirs, Vol. 1, 1936.Google Scholar
  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.Google Scholar
  5. 5.
    Dantzig, G.,Linear Programming and Extensions, Princeton University Press, Princeton, New Jersey, 1963.Google Scholar
  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.Google Scholar
  7. 7.
    Witzgall, C.,Optimal Location of a Central Facility: Mathematical Models and Concepts, National Bureau of Standards, Report No. 8388, 1965.Google Scholar
  8. 8.
    Bellman, R.,An Application of Dynamic Programming to Location-Allocation Problems, SIAM Review, Vol. 7, No. 1, 1965.Google Scholar
  9. 9.
    Francis, R.,Some Aspects of a Minimax Location Problem, Operations Research, Vol. 15, No. 6, 1967.Google Scholar
  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.Google Scholar
  11. 11.
    Kadane, J.,Discrete Search and the Neyman-Pearson Lemma, Journal of Mathematical Analysis and Applications, Vol. 22, No. 1, 1968.Google Scholar
  12. 12.
    Lehmann, E.,Testing Statistical Hypotheses, John Wiley and Sons, New York, 1959.Google Scholar
  13. 13.
    Dantzig, G., andWald, A.,On the Fundamental Lemma of Neyman and Pearson, Annals of Mathematical Statistics, Vol. 22. No. 1, 1951.Google Scholar
  14. 14.
    Chernoff, H., andScheffé, H.,A Generalization of the Neyman-Pearson Fundamental Lemma, Annals of Mathematical Statistics, Vol. 23, No. 2, 1952.Google Scholar
  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.Google Scholar
  16. 16.
    Virsan, C.,The Neyman-Pearson Lemma and Linear Programming, Revue Roumaine de Mathématiques Pures et Appliquées, Vol. 12, No. 2, 1967.Google Scholar
  17. 17.
    Zahl, S.,An Allocation Problem with Applications to Operations Research and Statistics, Operations Research, Vol. 11, No. 3, 1963.Google Scholar
  18. 18.
    Capon, J.,On the Asymptotic Efficiency of Locally Optimum Detectors, IRE Transactions on Information Theory, Vol. 7, No. 2, 1961.Google Scholar
  19. 19.
    Francis, R.,Sufficient Conditions for Some Optimum-Property Facility Designs, Operations Research, Vol. 15, No. 3, 1967.Google Scholar
  20. 20.
    Reinhardt, H. E.,A Maximization Problem Suggested by Baker versus Carr, American Mathematical Monthly, Vol. 73, No. 10, 1966.Google Scholar
  21. 21.
    Wagner, D. H.,Nonlinear Functional Versions of the Neyman-Pearson Lemma, SIAM Review, Vol. 11, No. 1, 1969.Google Scholar
  22. 22.
    Beltrami, E. J.,A Penalty Approach to a Variational Lemma Concerning Optimum Allocation (to appear).Google Scholar
  23. 23.
    Eggleston, H. G.,Convexity, Cambridge University Press, Cambridge, England, 1958.Google Scholar
  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.Google Scholar
  25. 25.
    Rockafellar, R. T.,Duality and Stability in Extremum Problems Involving Convex Functions, Pacific Journal of Mathematics, Vol. 22, No. 1, 1967.Google Scholar
  26. 26.
    Karlin, S.,Mathematical Methods and Theory in Games, Programming and Economics, Vol. 2, Addison-Wesley Publishing Company, Reading, Massachusetts, 1959.Google Scholar
  27. 27.
    Royden, H. L.,Real Analysis, 2nd edition, The Macmillan Company, New York, 1968.Google Scholar
  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.Google Scholar

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.Google Scholar
  2. 30.
    Neyman, J., andPearson, E.,Joint Statistical Papers, University of California Press, Berkeley, California, 1967.Google Scholar
  3. 31.
    Nering, E. D.,Linear Algebra and Matrix Theory, John Wiley and Sons, New York, 1967.Google Scholar
  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.Google Scholar

Copyright information

© Plenum Publishing Corporation 1969

Authors and Affiliations

  • Richard L. Francis
    • 1
  • Gordon P. Wright
    • 2
  1. 1.Department of Industrial EngineeringThe Ohio State UniversityColumbus
  2. 2.Department of Industrial Engineering and Management SciencesNorthwestern UniversityEvanston

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