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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

Abstract

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.

Keywords

Nonlinear programming Lagrange multiplier theorem First-order necessary conditions 

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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

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