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.
