Hessian Matrix - Bordered Hessian

A bordered Hessian is used for the second-derivative test in certain constrained optimization problems. Given the function as before:

but adding a constraint function such that:

the bordered Hessian appears as

H(f,g) = \begin{bmatrix} 0 & \dfrac{\partial g}{\partial x_1} & \dfrac{\partial g}{\partial x_2} & \cdots & \dfrac{\partial g}{\partial x_n} \\ \dfrac{\partial g}{\partial x_1} & \dfrac{\partial^2 f}{\partial x_1^2} & \dfrac{\partial^2 f}{\partial x_1\,\partial x_2} & \cdots & \dfrac{\partial^2 f}{\partial x_1\,\partial x_n} \\ \dfrac{\partial g}{\partial x_2} & \dfrac{\partial^2 f}{\partial x_2\,\partial x_1} & \dfrac{\partial^2 f}{\partial x_2^2} & \cdots & \dfrac{\partial^2 f}{\partial x_2\,\partial x_n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ \dfrac{\partial g}{\partial x_n} & \dfrac{\partial^2 f}{\partial x_n\,\partial x_1} & \dfrac{\partial^2 f}{\partial x_n\,\partial x_2} & \cdots & \dfrac{\partial^2 f}{\partial x_n^2} \end{bmatrix}

If there are, say, m constraints then the zero in the north-west corner is an m × m block of zeroes, and there are m border rows at the top and m border columns at the left.

The above rules stating that extrema are characterized by a positive definite or negative definite Hessian cannot apply here since a bordered Hessian cannot be definite: we have z'Hz = 0 if vector z has a non-zero as its first element, followed by zeroes.

The second derivative test consists here of sign restrictions of the determinants of a certain set of n - m submatrices of the bordered Hessian. Intuitively, one can think of the m constraints as reducing the problem to one with n - m free variables. (For example, the maximization of subject to the constraint can be reduced to the maximization of without constraint.)

Specifically, sign conditions are imposed on the sequence of principal minors (determinants of upper-left-justified sub-matrices) of the bordered Hessian, the smallest minor consisting of the truncated first 2m+1 rows and columns, the next consisting of the truncated first 2m+2 rows and columns, and so on, with the last being the entire bordered Hessian. There are thus n–m minors to consider. A sufficient condition for a local maximum is that these minors alternate in sign with the smallest one having the sign of (–1)m+1. A sufficient condition for a local minimum is that all of these minors have the sign of (–1)m. (Note that in the unconstrained case of m=0 these conditions coincide with the conditions for the unbordered Hessian to be negative definite or positive definite respectively.)

Read more about this topic:  Hessian Matrix