Optimization Letters

, Volume 6, Issue 8, pp 1597–1601 | Cite as

A short elementary proof of the Lagrange multiplier theorem

  • Olga BrezhnevaEmail author
  • Alexey A. Tret’yakov
  • Stephen E. Wright
Original Paper


We present a short elementary proof of the Lagrange multiplier theorem for equality-constrained optimization. Most proofs in the literature rely on advanced analysis concepts such as the implicit function theorem, whereas elementary proofs tend to be long and involved. By contrast, our proof uses only basic facts from linear algebra, the definition of differentiability, the critical-point condition for unconstrained minima, and the fact that a continuous function attains its minimum over a closed ball.


Nonlinear programming Lagrange multiplier theorem First-order necessary conditions 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Beltrami E.J.: A constructive proof of the Kuhn-Tucker multiplier rule. J. Math. Anal. Appl. 26, 297–306 (1969)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Bertsekas D.P.: Nonlinear programming. Athena Scientific, Boston (1995)zbMATHGoogle Scholar
  3. 3.
    Birbil S.I., Frenk J.B.G., Still G.J.: An elementary proof of the Fritz–John and Karush–Kuhn–Tucker conditions in nonlinear programming. Euro. J. Oper. Res. 180, 479–484 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Brezhneva O.A., Tret’yakov A.A., Wright S.E.: A simple and elementary proof of the Karush–Kuhn–Tucker theorem for inequality-constrained optimization. Optim. Lett. 3, 7–10 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Clarke F.H.: Optimization and nonsmooth analysis. Wiley, New York (1983)zbMATHGoogle Scholar
  6. 6.
    Craven B.D.: Mathematical programming and control theory. Chapman and Hall, London (1978)zbMATHCrossRefGoogle Scholar
  7. 7.
    Halkin H.: Implicit functions and optimization problems without continuous differentiability of the data. SIAM J. Control 12, 229–236 (1974)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Luenberger D.G.: Introduction to linear and nonlinear programming. Addison-Wesley, Reading (1973)zbMATHGoogle Scholar
  9. 9.
    McShane E.J.: The Lagrange multiplier rule. Amer. Math. Monthly 80, 922–925 (1973)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Pourciau B.H.: Modern multiplier rules. Amer. Math. Monthly 87, 433–452 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Rockafellar R.T.: Lagrange multipliers and optimality. SIAM Rev. 35, 183–238 (1993)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  • Olga Brezhneva
    • 1
    Email author
  • Alexey A. Tret’yakov
    • 2
    • 3
    • 4
  • Stephen E. Wright
    • 5
  1. 1.Department of MathematicsMiami UniversityOxfordUSA
  2. 2.Dorodnicyn Computing Center of the Russian Academy of SciencesMoscowRussia
  3. 3.System Research InstitutePolish Academy of SciencesWarsawPoland
  4. 4.University of Podlasie in SiedlceSiedlcePoland
  5. 5.Department of StatisticsMiami UniversityOxfordUSA

Personalised recommendations