In mathematics, the Hessian matrix is the square matrix of second-order partial derivatives of a function. Given the real-valued function
- <math>f(x_1, x_2, \dots, x_n),\,\!</math>
if all second partial derivatives of f exist, then the Hessian matrix of f is the matrix
- <math>H(f)_{ij}(x) = D_i D_j f(x)\,\!</math>
where x = (x1, x2, …, xn) and Di is the differentiation operator with respect to the ith argument:
- <math>H(f) = \begin{bmatrix}
\frac{\partial^2 f}{\partial x_1^2} & \frac{\partial^2 f}{\partial x_1\,\partial x_2} & \cdots & \frac{\partial^2 f}{\partial x_1\,\partial x_n} \\ \\
\frac{\partial^2 f}{\partial x_2\,\partial x_1} & \frac{\partial^2 f}{\partial x_2^2} & \cdots & \frac{\partial^2 f}{\partial x_2\,\partial x_n} \\ \\
\vdots & \vdots & \ddots & \vdots \\ \\
\frac{\partial^2 f}{\partial x_n\,\partial x_1} & \frac{\partial^2 f}{\partial x_n\,\partial x_2} & \cdots & \frac{\partial^2 f}{\partial x_n^2}
\end{bmatrix}</math>
(some workers define the Hessian as the determinant of the above matrix). The term “Hessian” was coined by James Joseph Sylvester, named for German mathematician Ludwig Otto Hesse, who had used the term “functional determinants”.
Hessian matrices are used in large-scale optimization problems within Newton-type methods. However, the full Hessian matrix can be difficult to compute in practice; in such situations, quasi-Newton algorithms have been developed that use approximations to the Hessian. The most well-known quasi-Newton algorithm is the BFGS algorithm.
Mixed derivatives and symmetry of the Hessian
The mixed derivatives of f are the entries off the main diagonal in the Hessian. Assuming that they are continuous, the order of differentiation does not matter (Clairaut’s theorem). For example,
- <math>\frac {\partial}{\partial x} \left( \frac { \partial f }{ \partial y} \right) =
\frac {\partial}{\partial y} \left( \frac { \partial f }{ \partial x} \right)</math>
This can also be written (in reverse order) as:
- <math>f_{xy} = f_{yx} \,</math>
In a formal statement: if the second derivatives of f are all continuous in a region D, then the Hessian of f is a symmetric matrix throughout D; see symmetry of second derivatives.
Critical points and discriminant
If the gradient of f (i.e. its derivative in the vector sense) is zero at some point x, then f has a critical point (or stationary point) at x. The determinant of the Hessian at x is then called the discriminant. If this determinant is zero then x is called a degenerate critical point of f, this is also called a non-Morse critical point of f. Otherwise it is non-degenerate, this is called a Morse critical point of f.
Second derivative test
The following test can be applied at a non-degenerate critical point x. If the Hessian is positive definite at x, then f attains a local minimum at x. If the Hessian is negative definite at x, then f attains a local maximum at x. If the Hessian has both positive and negative eigenvalues then x is a saddle point for f (this is true even if x is degenerate). Otherwise the test is inconclusive.
Note that for positive semidefinite and negative semidefinite Hessians the test is inconclusive. However, more can be said from the point of view of Morse theory.
In view of what has just been said, the second derivative test for functions of one and two variables is simple. In one variable, the Hessian contains just one second derivative; if it is positive then x is a local minimum, if it is negative then x is a local maximum; if it is zero then the test is inconclusive. In two variables, the discriminant can be used, because the determinant is the product of the eigenvalues. If it is positive then the eigenvalues are both positive, or both negative. If it is negative then the two eigenvalues have different signs. If it is zero, then the second derivative test is inconclusive.
Bordered Hessian
A bordered Hessian is used for the second-derivative test in certain constrained optimization problems. Given the function as before:
- <math>f(x_1, x_2, \dots, x_n),</math>
but adding a constraint function such that:
- <math>g(x_1, x_2, \dots, x_n) = c,</math>
the bordered Hessian appears as
- <math>H(f,g) = \begin{bmatrix}
0 & \frac{\partial g}{\partial x_1} & \frac{\partial g}{\partial x_2} & \cdots & \frac{\partial g}{\partial x_n} \\ \\
\frac{\partial g}{\partial x_1} & \frac{\partial^2 f}{\partial x_1^2} & \frac{\partial^2 f}{\partial x_1\,\partial x_2} & \cdots & \frac{\partial^2 f}{\partial x_1\,\partial x_n} \\ \\
\frac{\partial g}{\partial x_2} & \frac{\partial^2 f}{\partial x_2\,\partial x_1} & \frac{\partial^2 f}{\partial^2 x_2} & \cdots & \frac{\partial^2 f}{\partial x_2\,\partial x_n} \\ \\
\vdots & \vdots & \vdots & \ddots & \vdots \\ \\
\frac{\partial g}{\partial x_n} & \frac{\partial^2 f}{\partial x_n\,\partial x_1} & \frac{\partial^2 f}{\partial x_n\,\partial x_2} & \cdots & \frac{\partial^2 f}{\partial x_n^2}
\end{bmatrix}.</math>
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 of positive definite and negative definite can not apply here since a bordered Hessian can not 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 HessianMagnus, J.R. and H. Neudecker: “Matrix Differential Calculus with Applications in Statistics and Econometrics”, page 136. Wiley, 1988. Intuitively, think of the m constraints as reducing the problem to one with n-m free variables. (For example, the maximization of <math>f(x_1,x_2,x_3)</math> subject to the constraint <math>x_1+x_2+x_3=1</math> can be reduced to the maximization of <math>f(x_1,x_2,1-x_1-x_2)</math> without constraint.)
Vector-valued functions
If f is instead vector-valued, i.e.
- <math>f = (f_1, f_2, \dots, f_n),</math>
then the array of second partial derivatives is not a matrix, but a tensor of rank 3.
See also
Notes