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# Lagrange Condition

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If the scalar function has a local extremum at a point subject to the constraints , then there exist Lagrange multipliers , such that

This characterization requires that and are continuously differentiable in a neighborhood of and that the Jacobi matrix has maximal rank.

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

if , i.e., the level sets of and touch at the extreme point.

The Lagrange condition is not sufficient to decide if 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 is not maximal.

Let denote the number of variables and the number of constraints.

For there is nothing to show since an arbitrary -vector can always be represented as linear combination of linear independent rows of .

For denote by a partition of the variables, where, after permutation, we can assume that is invertible. Then, by the implicit function theorem, the constraints can locally be solved for in terms of :

As a consequence, the gradient of the function vanishes at an extremum:

Moreover, differentiating the contraints , it follows that

Substituting into the expression for the gradient, we see that, with

the equations

are valid at the point . These identities correspond to the - and -components of the Lagrange condition .

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 automatisch erstellt am 26.  1. 2017