A short elementary proof of the Lagrange multiplier theorem

作者：Brezhneva Olga^*; Tret' yakov Alexey A; Wright Stephen E

来源：Optimization Letters, 2012, 6(8): 1597-1601.

DOI：10.1007/s11590-011-0349-4

• Nonlinear programming
• Lagrange multiplier theorem
• First-order necessary conditions

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.

• 出版日期2012-12