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## Lagrange Condition |

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z | overview |

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

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

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 :

Moreover, differentiating the contraints , it follows that

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

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automatisch erstellt am 26. 1. 2017 |