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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.

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