Modified Hessian matrix

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). ${\displaystyle k>2m}$ ${\displaystyle k>2\cdot 1\rightarrow k>2}$

Be open The function is twice continuously differentiable and it has a local extreme under the constraints with . Be now ${\displaystyle U\subset \mathbb {R} ^{n}}$ ${\displaystyle f:U\rightarrow \mathbb {R} }$ ${\displaystyle a\in U}$ ${\displaystyle F=(F_{1},\ldots ,\,F_{m}):U\rightarrow \mathbb {R} ^{m}}$ ${\displaystyle m<n}$

${\displaystyle L(\lambda _{1},\ldots ,\,\lambda _{m},\,x):=f(x)-\sum _{i=1}^{m}\lambda _{i}F_{i}(x)}$

the Lagrange function, which is short for . Then the rimmed Hessian matrix is ​​understood to be the matrix ${\displaystyle x}$ ${\displaystyle x_{1},\ldots ,\,x_{n}}$ ${\displaystyle (n+m)\times (n+m)}$

${\displaystyle {\begin{aligned}\operatorname {\overline {H}} ({\bar {\lambda }},\,a)&:=\left({\begin{array}{cccccc}{\frac {\partial ^{2}L}{\partial \lambda _{1}^{2}}}&\ldots &{\frac {\partial ^{2}L}{\partial \lambda _{1}\partial \lambda _{m}}}&{\frac {\partial ^{2}L}{\partial \lambda _{1}\partial x_{1}}}&\ldots &{\frac {\partial ^{2}L}{\partial \lambda _{1}\partial x_{n}}}\\\vdots &\ddots &\vdots &\vdots &&\vdots \\{\frac {\partial ^{2}L}{\partial \lambda _{m}\partial \lambda _{1}}}&\ldots &{\frac {\partial ^{2}L}{\partial \lambda _{m}^{2}}}&{\frac {\partial ^{2}L}{\partial \lambda _{m}\partial x_{1}}}&\ldots &{\frac {\partial ^{2}L}{\partial \lambda _{m}\partial x_{n}}}\\{\frac {\partial ^{2}L}{\partial x_{1}\partial \lambda _{1}}}&\ldots &{\frac {\partial ^{2}L}{\partial x_{1}\partial \lambda _{m}}}&{\frac {\partial ^{2}L}{\partial x_{1}^{2}}}&\ldots &{\frac {\partial ^{2}L}{\partial x_{1}\partial x_{n}}}\\\vdots &&\vdots &\vdots &\ddots &\vdots \\{\frac {\partial ^{2}L}{\partial x_{n}\partial \lambda _{1}}}&\ldots &{\frac {\partial ^{2}L}{\partial x_{n}\partial \lambda _{m}}}&{\frac {\partial ^{2}L}{\partial x_{n}\partial x_{1}}}&\ldots &{\frac {\partial ^{2}L}{\partial x_{n}^{2}}}\end{array}}\right)({\bar {\lambda }},\,a)\\\end{aligned}}}$

or already simplified

${\displaystyle {\begin{aligned}\operatorname {\overline {H}} ({\bar {\lambda }},\,a)&:=\left({\begin{array}{cccccc}0&\ldots &0&-{\frac {\partial F_{1}}{\partial x_{1}}}&\ldots &-{\frac {\partial F_{1}}{\partial x_{n}}}\\\vdots &\ddots &\vdots &\vdots &&\vdots \\0&\ldots &0&-{\frac {\partial F_{m}}{\partial x_{1}}}&\ldots &-{\frac {\partial F_{m}}{\partial x_{n}}}\\-{\frac {\partial F_{1}}{\partial x_{1}}}&\ldots &-{\frac {\partial F_{m}}{\partial x_{1}}}&{\frac {\partial ^{2}L}{\partial x_{1}^{2}}}&\ldots &{\frac {\partial ^{2}L}{\partial x_{1}\partial x_{n}}}\\\vdots &&\vdots &\vdots &\ddots &\vdots \\-{\frac {\partial F_{1}}{\partial x_{n}}}&\ldots &-{\frac {\partial F_{m}}{\partial x_{n}}}&{\frac {\partial ^{2}L}{\partial x_{n}\partial x_{1}}}&\ldots &{\frac {\partial ^{2}L}{\partial x_{n}^{2}}}\end{array}}\right)({\bar {\lambda }},\,a)\end{aligned}}}$

with the associated solutions of the auxiliary variables. ${\displaystyle {\bar {\lambda }}={\bar {\lambda }}_{1},\ldots ,\,{\bar {\lambda }}_{m}}$

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. ${\displaystyle L(x_{1},x_{2})=f(x_{1},x_{2})+\lambda g(x_{1},x_{2})}$ ${\displaystyle f:\mathbb {R} ^{2}\rightarrow \mathbb {R} ,(x_{1},x_{2})\mapsto f(x_{1},x_{2})}$ ${\displaystyle g(x_{1},x_{2})=0\,}$

${\displaystyle \operatorname {\bar {H}} (x)={\begin{pmatrix}0&g_{x1}&g_{x2}\\g_{x1}&L_{x1x1}&L_{x1x2}\\g_{x2}&L_{x2x1}&L_{x2x2}\\\end{pmatrix}}={\begin{pmatrix}0&{\frac {\partial g}{\partial x_{1}}}&{\frac {\partial g}{\partial x_{2}}}\\[1.5ex]{\frac {\partial g}{\partial x_{1}}}&{\frac {\partial ^{2}L}{\partial x_{1}^{2}}}&{\frac {\partial ^{2}L}{\partial x_{1}\partial x_{2}}}\\[1.5ex]{\frac {\partial g}{\partial x_{2}}}&{\frac {\partial ^{2}L}{\partial x_{2}\partial x_{1}}}&{\frac {\partial ^{2}L}{\partial x_{2}^{2}}}\\\end{pmatrix}}}$

The one at the top left in the matrix comes about through . ${\displaystyle 0}$ ${\displaystyle \operatorname {\bar {H}} _{11}={\frac {\partial ^{2}L}{\partial \lambda ^{2}}}}$

A stationary position of is then subject to the secondary condition ${\displaystyle x_{0}}$ ${\displaystyle f}$ ${\displaystyle g}$

• local maximum if ${\displaystyle \det {\bar {H}}(x_{0})>0}$
• local minimum if ${\displaystyle \det {\bar {H}}(x_{0})<0}$
• undecidable if ${\displaystyle \det {\bar {H}}(x_{0})=0}$

Web links

• Modified Hessian matrix. University of Economy Vienna.
• Robert Koschig: The optimization method with Lagrange multipliers.