Optimization and Lagrange multipliers: non-C1 constraints and “minimal” constraint qualifications

  • Leonid Hurwicz
  • Marcel K. Richter
Part of the Advances in Mathematical Economics book series (MATHECON, volume 5)


Constrained optimization problems are central to economics, and Lagrange multipliers — when they exist — play a basic role in solving them, in theory and in practice. Examples are well known of optimization problems for which multipliers do not exist. So it is important to know what requirements constraint functions must satisfy to be “Lagrange regular,” i.e. to guarantee existence of multipliers for broad classes of maximand or minimand functions. We relax the requirements in three directions:
  1. 1

    We reduce the smoothness requirements on constraints. This allows weaker and more uniform hypotheses for mixed inequality and equality constraints, permitting, for example, just differentiability at the optimum and continuity in a neighborhood. (We allow much weaker hypotheses, as well.) Beyond smoothness, other requirements have long been imposed on constraint functions, to avoid simple examples lacking multipliers. We examine two types of such “constraint qualifications”.

  2. 2

    We provide new, relaxed constraint qualifications of both Jacobian and path types.



Partial Derivative Lagrange Multiplier Equality Constraint Inequality Constraint Path Criterion 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abadie, J.: On the Kuhn-Tucker Theorem. In: Nonlinear Programming, North-Holland, New York 1967Google Scholar
  2. 2.
    Arrow, K.J., Hurwicz, L.: Reduction of constrained maxima to saddle-point problems. In: Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, volume V, pp. 1–20 University of California Press, Berkeley and Los Angeles 1956Google Scholar
  3. 3.
    Arrow, K.J., Hurwicz, L. eds.: Studies in Resource Allocation Processes. Cambridge University Press, Cambridge 1978Google Scholar
  4. 4.
    Arrow, K.J., Hurwicz, L., Uzawa, H.: Constraint qualifications in maximization problems. Naval Research Logistics Quarterly 8, 175–191 (1961)CrossRefGoogle Scholar
  5. 5.
    Bliss, G.A.: Normality and abnormality in the calculus of variations. Transactions of the American Mathematical Society 43, 365–376 (1938)CrossRefGoogle Scholar
  6. 6.
    Bliss, G.A.: Lectures on the Calculus of Variations. University of Chicago Press, Chicago 1946Google Scholar
  7. 7.
    Bolza, O.: Vorlesungen über Variationsrechnung. Chelsea Publishing Company, New York (second edition, no date, author’s preface dated 1909)Google Scholar
  8. 8.
    Bouligand, G.: Introduction à la Géométrie Infinitésimale Directe. Librairie Vuibert, Paris 1932Google Scholar
  9. 9.
    Carathéodory, C: Calculus of Variations and Partial Differential Equations of the First Order. Chelsea Publishing Company, New York 1982 (Second (revised) edition originally published as Variationsrechnung und Partielle Differentialgleichungen erster Ordnung, B.G.Teubner, Berlin 1935)Google Scholar
  10. 10.
    de la Grange, J.L.: Méchanique Analitique. Chez la Veuve Desaint, Libraire, Paris, 1788 (See also Lagrange, J.L.)Google Scholar
  11. 11.
    El-Hodiri, M.A.: The Karush characterization of constrained extrema of functions of a finite number of variables. Series A, No. 3, UAR Ministry of Treasury Research Memoranda, July 1967Google Scholar
  12. 12.
    Euler, L.: Methodus Inveniendi Lineas Curvas. In: Leonhardi Euleri Opera Omnia, series prima XXIV, vol. 34. Swiss Society of Natural Sciences, Bern 1952 (Reprinted from Methodus Inveniendi Lineas Curvas, Lausanne Geneva 1744)Google Scholar
  13. 13.
    Evans, J.P: On constraint qualifications in nonlinear programming. Report 6917, Center for Mathematical Studies in Business and Economics, Graduate School of Business, University of Chicago, Chicago, May 1969Google Scholar
  14. 14.
    Goldstine, H.H.: A History of the Calculus of Variations. Springer-Verlag, New York 1980Google Scholar
  15. 15.
    Gould, F.J., Tolle, J.W.: A necessary and sufficient qualification for constrained optimization. SIAM Journal of Applied Mathematics 20, 164–172 (1971)CrossRefGoogle Scholar
  16. 16.
    Halkin, H.: Implicit functions and optimization problems without continuous differentiability of the data. SIAM Journal on Control 12, 229–236 (1974)CrossRefGoogle Scholar
  17. 17.
    Hancock, H.: Theory of Maxima and Minima. Ginn and Company, Boston 1917Google Scholar
  18. 18.
    Hestenes, M.R.: Calculus of Variations and Optimal Control Theory. John Wiley & Sons, New York 1966Google Scholar
  19. 19.
    Hestenes, M.R.: Optimization Theory. John Wiley & Sons, New York 1975Google Scholar
  20. 20.
    Hurwicz, L.: Programming in Linear Spaces. In: Studies in Linear and Non-Linear Programming (Kenneth J. Arrow, Leonid Hurwicz, Hirofumi Uzawa eds.). Chapter 4, pp.38–102 Stanford University Press, Stanford 1958Google Scholar
  21. 21.
    Hurwicz, L., Richter, M.K.: Implicit Functions and Diffeomorphisms without C 1. Discussion Paper No. 279, Department of Economics, University of Minnesota 1994 (see also this volume, pp.65–96)Google Scholar
  22. 22.
    Hurwicz, L., Richter, M.K.: Optimization and Lagrange Multipliers: Non-C 1 Constraints and “Minimal” Constraint Qualifications. Discussion Paper No. 280, Department of Economics, University of Minnesota 1995Google Scholar
  23. 23.
    Ioffe, A.D., Tihomirov, V.M.: Theory of Extremal Problems. North-Holland, Amsterdam 1979Google Scholar
  24. 24.
    John, F.: Extremum problems with inequalities as subsidiary conditions. In: Studies and Essays: Courant Anniversary Volume (K. O. Friedrichs, O. E. Neugebauer, J. J. Stoker eds.). pp. 187–204 Interscience Publishers, New York 1948Google Scholar
  25. 25.
    Karush, W.: Minima of functions of several variables with inequalities as side conditions. Master’s dissertation 1939Google Scholar
  26. 26.
    Klee, V.L. Jr.: Separation properties of convex cones. Proceedings of the American Mathematical Society 6, 313–318 (1955)CrossRefGoogle Scholar
  27. 27.
    Kneser, A.: Variationsrechnung. Teubner, Leipzig, 1899–1916 (Originally published as Heft 5 1904)Google Scholar
  28. 28.
    Kuhn, H.W.: Solvability and consistency for linear equations and inequalities. American Mathematical Monthly 63, 217–232 (1956)CrossRefGoogle Scholar
  29. 29.
    Kuhn, H.W., Tucker, A.W.: Nonlinear programming. In: Second Berkeley Symposium on Mathematical Statistics and Probability (J. Neyman ed.). pp.481–492 University of California Press 1951Google Scholar
  30. 30.
    Kuhn, H.W.: Nonlinear Programming: A Historical View. In: Nonlinear Programming (Richard W. Cottle, C. E. Lemke eds.). American Mathematical Society 1976Google Scholar
  31. 31.
    Lagrange, J.L.: Théorie des Fonctions Analytiques. Courcier, Paris 1813, Nouvelle édition. See also de la Grange, J.L.)Google Scholar
  32. 32.
    Mangasarian, O.L., Fromovitz, S.: The Fritz John necessary optimality conditions in the presence of equality and inequality constraints. Journal of Mathematical Analysis and Applications 17, 37–47 (1967)CrossRefGoogle Scholar
  33. 33.
    Mangasarian, O.L.: Nonlinear Programming. McGraw-Hill, New York 1969Google Scholar
  34. 34.
    McFadden, D., Richter, M.K.: Stochastic rationality and revealed stochastic preferences. In: Preferences, Uncertainty, and Optimality: Essays in Honor of Leonid Hurwicz (John S. Chipman, Daniel McFadden and Marcel K. Richter eds.). chapter 6. Westview Press, Boulder, CO 1990Google Scholar
  35. 35.
    Motzkin, T.S.: Beiträge zur Theorie der linearen Ungleichungen. Dissertation, Basle 1934Google Scholar
  36. 36.
    Pennisi, L.L.: An indirect sufficiency proof for the problem of Lagrange with differential inequalities as added side conditions. Transactions of the American Mathematical Society 74, 177–198 (1953)CrossRefGoogle Scholar
  37. 37.
    Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton 1970Google Scholar
  38. 38.
    Slater, M.: Lagrange Multipliers Revisited: A Contribution to Nonlinear Programming. Cowles Commission Discussion Paper, Math. 403, November 1950Google Scholar
  39. 39.
    Stoer, J., Witzgall, C: Convexity and Optimization in Finite Dimensions, volume I. Springer-Verlag, New York 1970Google Scholar
  40. 40.
    Takayama, A.: Mathematical Economics. The Dry den Press, Hinsdale, IL 1974Google Scholar
  41. 41.
    Valentine, F.A.: The problem of Lagrange with differential inequalities as added side conditions. In: Contributions to the Calculus of Variations 1933-1937: Theses Submitted to the Department of Mathematics of the University of Chicago, pp.403–447. University of Chicago Press, Chicago 1937Google Scholar
  42. 42.
    Varaiya P.P.: Nonlinear programming in Banach space. SIAM Journal of Applied Mathematics 15, 284–293 (1967)CrossRefGoogle Scholar
  43. 43.
    Weierstrass, K.: Mathematische Werke. Akademische Verlagsgesellschaft, Leipzig 1927 (Edited from notes of Weierstrass’s 1875-1878 lectures.)Google Scholar

Copyright information

© Springer-Verlag Tokyo 2003

Authors and Affiliations

  • Leonid Hurwicz
    • 1
  • Marcel K. Richter
    • 1
  1. 1.Department of EconomicsUniversity of MinnesotaMinneapolisUSA

Personalised recommendations