An elementary proof of the Lagrange multiplier theorem in normed linear spaces

Olga Brezhneva, Alexey A. Tret'yakov
2012 Optimization  
We present an elementary proof of the Lagrange multiplier theorem for optimization problems with equality constraints in normed linear spaces. Most proofs in the literature rely on advanced concepts and results, such as the implicit function theorem and the Lyusternik theorem. By contrast, the proof given in this article employs only basic results from linear algebra, the critical-point condition for unconstrained minima and the fact that a continuous function attains its minimum over a closed ball in the finite-dimensional space.
doi:10.1080/02331934.2011.603323 fatcat:oz56kesvlze7ta3gli4ykkjp3e