how can we calculate the hessian matrix in general ?

is there any clear way for calculating the hessian matrix of a complex equation? for example in QP problems we have to change the form of our equation to form of quadprog() function. how can we calculate the matrix H and f in quadprog in general. i asked similar question before in this link: http://www.mathworks.com/matlabcentral/answers/234018-please-solving-this-qp-equation but i cant understand the answers ... please answer my question in detail

Answer 1

[Matt J]

Well, the Hessian of a function g(x) is by definition the matrix of second partial derivatives

   H(i,j) = d^2/(dxi dxj) g(x)

so it can always be calculated that way. As for f, when the objective g(x) is quadratic, f is the gradient of g at x=0 and can likewise be calculated by directly taking partial derivatives.

However, often you don't have to resort to these basic definitions to compute the Hessian. You can put the terms of your function in familiar forms whose Hessian is very easy to inspect. If you haven't done so, for example, you should convince yourself that the Hessian of the general quadratic form

    g(x) = (x.' * Q * x)/2

is H=Q. You would never go through the labor of taking partial derivatives for something so basic.

Accordingly, your main problem in your previous thread is that you didn't put your function in matrix-vector form. You have all the terms in your function expanded into sums like,

   sum_i sum_j alpha_i*alpha_j*(x_i.' * x_j)

As I explained to you then, the expression above can be written in matrix-vector form as

    alpha.'*(X.'*X)*alpha

from which it is easy to see that the Hessian of this term is H=2*(X.'*X). The other terms become similarly easy to manage when put in matrix-vector form.

[Hassan]

matt thank you ... your answer is very complete and in detail ... i got it now what you do in my previous question ... :-)

[sonu Lamba]

Hello everyone, Can you tell how to calculate hessian matrix for a spatio-temporal video sequence I(x,y,t)?. Here, t corresponds to next frame. I need to calculate hessian matrix in videos for keypoints detection

Answer 2

[Dauren]

I think answer to you question is hessian function.

If you have function y=x1^2+2*x1*x2+x2^2

and you want to represent y in form of y=x*H*x',

where x=[x1 x2],

then apply:

syms x1 x2
y=x1^2+2*x1*x2+x2^2
H=1/2.*hessian(y)

[Matt J]

Note that this won't be practical with a large number (e.g., 100, 1000) of unknown x(i). The Symbolic Math Toolbox unfortunately does not support symbolic manipulation of vector variables.