Modified Hessian matrix

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The rimmed Hessian matrix (engl. Bordered Hessian ) is used for classification of stationary points in the multidimensional extreme problems with constraints. It is related to the "normal" Hesse matrix . In contrast to the Hessian matrix, which is examined for positive or negative definiteness , the sign of the determinant is decisive for the modified Hessian matrix .

The decisive factor is the sequence of signs of the leading major minors , whereby it applies that only the k leading major minors are examined for which the following applies: (m number of constraints) For example, if you examine a function for variables with a secondary condition, you have to consider, i.e. first the signs from the 3rd leading major minor (see also the following example).

Be open The function is twice continuously differentiable and it has a local extreme under the constraints with . Be now

the Lagrange function, which is short for . Then the rimmed Hessian matrix is ​​understood to be the matrix

or already simplified

with the associated solutions of the auxiliary variables.

Shape (2-dimensional case)

For a two-dimensional function with a secondary condition, the modified Hessian matrix has the following form.

Let be the Lagrangian , where is any two-dimensional function and the constraint under which optimization should be performed.

The one at the top left in the matrix comes about through .

A stationary position of is then subject to the secondary condition

  • local maximum if
  • local minimum if
  • undecidable if

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