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.
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Brezhneva, O., Tret’yakov, A.A. & Wright, S.E. A short elementary proof of the Lagrange multiplier theorem. Optim Lett 6, 1597–1601 (2012). https://doi.org/10.1007/s11590-011-0349-4