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Lagrange Condition

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If the scalar function $ f$ has a local extremum at a point $ x_\star$ subject to the constraints $ g_i(x)=0$, then there exist Lagrange multipliers $ \lambda_i$, such that

$\displaystyle f^\prime(x_\star) = \lambda^{\operatorname t}g^\prime(x_\star) \,. $

This characterization requires that $ f$ and $ g$ are continuously differentiable in a neighborhood of $ x_\star$ and that the Jacobi matrix $ g^\prime(x_\star)$ has maximal rank.

For a single constraint, the Lagrange condition has the simple form

$\displaystyle \operatorname{grad} f(x_\star) \,\parallel\, \operatorname{grad} g(x_\star) \,, $

if $ \operatorname{grad} g(x_\star)\neq 0$, i.e., the level sets of $ f$ and $ g$ touch at the extreme point.

The Lagrange condition is not sufficient to decide if $ x_*$ is an extremum, or to determine its type. This requires additional information.

The global extrema of a function can be obtained by comparing the function values at the points which satisfy the Lagrange condition, the points on the boundary of the admissible set, and points where the rank of $ g^\prime$ is not maximal.

see also:

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  automatically generated 1/26/2017