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MATH
[1251]

Theorems from Constrained extrema

All theorems from Constrained extrema.

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Lagrange multipliers

Suppose C={(x,y):g(x,y)=0}C=\{(x,y):g(x,y)=0\} is a nonempty closed and bounded curve and ff and gg have continuous first partial derivatives on an open set containing CC. Then ff attains a maximum and a minimum on CC. At each constrained extremum where

∇g(x,y)≠0,\nabla g(x,y)\ne\mathbf{0},

there is a scalar λ∈R\lambda\in\mathbb{R} such that

∇f(x,y)=λ∇g(x,y){\nabla f(x,y)=\lambda\nabla g(x,y)}

together with

g(x,y)=0.{g(x,y)=0}.

The equation says that the gradients are parallel, not that they have equal length: λ\lambda records a scale and possibly a reversal of direction.

The regularity condition ∇g≠0\nabla g\ne\mathbf{0} is essential. A feasible point where ∇g=0\nabla g=\mathbf{0} is a singular constraint point, where the multiplier condition is not guaranteed, so it must be checked separately along with any corners or endpoints introduced by an inequality constraint. Compactness and continuity still guarantee that the extrema exist.

  1. [1]:Constrained extremum
  2. [2]:Gradient normal to a level set
  3. [3]:Proof
  4. [4]:Optimise subject to one equality constraint

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Lagrange multipliers