Hessian Matrix (Haisen matrix)

Hessian matrix, translated black plug matrices, Haisen matrices, Heather Matrix, sea plug matrix and so on. is a square of the second derivative of a multivariate function, and describes the local curvature of the function. The Hessian matrix was first proposed by the German mathematician Ludwig Otto Hesse in 19th century and named after its name.

Hessian matrix is commonly used to solve the optimization problem by Newton Method , and the Hessian matrix can determine the extremum problem of multivariate function.

In the optimization design of engineering practical problems, the listed objective functions are often very complex, in order to simplify the problem, the objective function is often expanded to approximate the original function at a certain point neighborhood, when the function expansions in the matrix form of Taylor expansion in some point will involve Hessian matrix.

Hessian Matrix of one or two-tuple functions

From the advanced mathematics knowledge, if the unary function f (x) has any derivative in a neighborhood of point X0, then f (x) at point X0 Taylor Expansion:

For the two-dollar function, the Taylor expansion at the point is:

The above is written in the form of a matrix:

The above-named abbreviations are:

which

It is the Hessian matrix at the point where the second derivative of the function at the point is formed.

Hessian Matrix of multivariate functions

The Taylor expansion of the two-ary function is generalized to the multivariate function, then the Taylor-expanded matrix at the point is:

which

, it is the gradient at the place.

For the Hessian Matrix at the place.

third, using Hessian matrix to determine the extremum of multivariate function

The n multivariate real function has a second-order continuous bias in the neighborhood of the point, if:

And

The

When a positive definite matrix is in place, the minimum value is

When a negative matrix is in place, the maximum value is

When a indeterminate matrix is not the extremum point

When a is a semi-positive definite matrix or a semi-negative definite matrix, it is a "suspicious" extremum point, which needs to be judged by other methods.