Editing Hessian matrix (section)

=== Bordered Hessian ===

A '''{{em|bordered Hessian}}''' is used for the second-derivative test in certain constrained optimization problems. Given the function <math>f</math> considered previously, but adding a constraint function <math>g</math> such that <math>g(\mathbf{x}) = c,</math> the bordered Hessian is the Hessian of the [[Lagrange multiplier|Lagrange function]] <math>\Lambda(\mathbf{x}, \lambda) = f(\mathbf{x}) + \lambda[g(\mathbf{x}) - c]</math>:<ref>{{cite web|title=Econ 500: Quantitative Methods in Economic Analysis I|date=October 7, 2004|first=Arne|last=Hallam|url=https://www2.econ.iastate.edu/classes/econ500/hallam/documents/opt_con_gen_000.pdf|work=Iowa State}}</ref>
<math display=block>\mathbf H(\Lambda) =
\begin{bmatrix}
\dfrac{\partial^2 \Lambda}{\partial \lambda^2} & \dfrac{\partial^2 \Lambda}{\partial \lambda \partial \mathbf x} \\
\left(\dfrac{\partial^2 \Lambda}{\partial \lambda \partial \mathbf x}\right)^{\mathsf{T}} & \dfrac{\partial^2 \Lambda}{\partial \mathbf x^2}
\end{bmatrix} =
\begin{bmatrix}
0 & \dfrac{\partial g}{\partial x_1} & \dfrac{\partial g}{\partial x_2} & \cdots & \dfrac{\partial g}{\partial x_n} \\[2.2ex]
\dfrac{\partial g}{\partial x_1} & \dfrac{\partial^2 \Lambda}{\partial x_1^2} & \dfrac{\partial^2 \Lambda}{\partial x_1\,\partial x_2} & \cdots & \dfrac{\partial^2 \Lambda}{\partial x_1\,\partial x_n} \\[2.2ex]
\dfrac{\partial g}{\partial x_2} & \dfrac{\partial^2 \Lambda}{\partial x_2\,\partial x_1} & \dfrac{\partial^2 \Lambda}{\partial x_2^2} & \cdots & \dfrac{\partial^2 \Lambda}{\partial x_2\,\partial x_n} \\[2.2ex]
\vdots & \vdots & \vdots & \ddots & \vdots \\[2.2ex]
\dfrac{\partial g}{\partial x_n} & \dfrac{\partial^2 \Lambda}{\partial x_n\,\partial x_1} & \dfrac{\partial^2 \Lambda}{\partial x_n\,\partial x_2} & \cdots & \dfrac{\partial^2 \Lambda}{\partial x_n^2}
\end{bmatrix}
= \begin{bmatrix}
0 & \dfrac{\partial g}{\partial \mathbf x} \\
\left(\dfrac{\partial g}{\partial \mathbf x}\right)^\mathsf{T} & \dfrac{\partial^2 \Lambda}{\partial \mathbf x^2}
\end{bmatrix}</math>

If there are, say, <math>m</math> constraints then the zero in the upper-left corner is an <math>m \times m</math> block of zeros, and there are <math>m</math> border rows at the top and <math>m</math> border columns at the left.

The above rules stating that extrema are characterized (among critical points with a non-singular Hessian) by a positive-definite or negative-definite Hessian cannot apply here since a bordered Hessian can neither be negative-definite nor positive-definite, as <math>\mathbf{z}^\mathsf{T} \mathbf{H} \mathbf{z} = 0</math> if <math>\mathbf{z}</math> is any vector whose sole non-zero entry is its first.

The second derivative test consists here of sign restrictions of the determinants of a certain set of <math>n - m</math> submatrices of the bordered Hessian.<ref>{{Cite book|last1=Neudecker|first1=Heinz|last2=Magnus|first2=Jan R.|title=Matrix Differential Calculus with Applications in Statistics and Econometrics|publisher=[[John Wiley & Sons]]|location=New York|isbn=978-0-471-91516-4|year=1988|page=136}}</ref> Intuitively, the <math>m</math> constraints can be thought of as reducing the problem to one with <math>n - m</math> free variables. (For example, the maximization of <math>f\left(x_1, x_2, x_3\right)</math> subject to the constraint <math>x_1 + x_2 + x_3 = 1</math> can be reduced to the maximization of <math>f\left(x_1, x_2, 1 - x_1 - x_2\right)</math> without constraint.)

Specifically, sign conditions are imposed on the sequence of leading principal minors (determinants of upper-left-justified sub-matrices) of the bordered Hessian, for which the first <math>2 m</math> leading principal minors are neglected, the smallest minor consisting of the truncated first <math>2 m + 1</math> rows and columns, the next consisting of the truncated first <math>2 m + 2</math> rows and columns, and so on, with the last being the entire bordered Hessian; if <math>2 m + 1</math> is larger than <math>n + m,</math> then the smallest leading principal minor is the Hessian itself.<ref>{{cite book|last=Chiang|first=Alpha C.|title=Fundamental Methods of Mathematical Economics|publisher=McGraw-Hill|edition=Third|year=1984|page=[https://archive.org/details/fundamentalmetho0000chia_b4p1/page/386 386]|isbn=978-0-07-010813-4|url=https://archive.org/details/fundamentalmetho0000chia_b4p1/page/386}}</ref> There are thus <math>n - m</math> minors to consider, each evaluated at the specific point being considered as a [[Candidate solution#Calculus|candidate maximum or minimum]]. A sufficient condition for a local {{em|maximum}} is that these minors alternate in sign with the smallest one having the sign of <math>(-1)^{m+1}.</math>  A sufficient condition for a local {{em|minimum}} is that all of these minors have the sign of <math>(-1)^m.</math> (In the unconstrained case of <math>m=0</math> these conditions coincide with the conditions for the unbordered Hessian to be negative definite or positive definite respectively).