How to calculate the second derivative of an unanalytically function

Dear All,

I use fmincon to optimize my likelihood function f (there are four parameters in f . And it is not in an analytic form. It needs to read in data and call other children function, etc). I also want the standard errors so I have tried to get the Hessian (more accurate, the diagonal part of it, which is the second derivative of the f at the optima). But Hessian provided by fmincon is the Hessian of the Lagrangian so I can not use it.

I tried John D'Errico's method . But in this method, it requires f to be scalar and analytic.

So I really have not idea at this moment. I think my problem is essentially about how to calculate the second derivative of an analytic function. Could anybody help me with this please? Thanks very much.

Xueqi

[Torsten]

The usual difference quotient does not help:

f''(x) ~ (f(x+h)-2*f(x)+f(x-h))/h^2

?

Best wishes

Torsten.

[xueqi]

Hi thanks for our advice. But I am not sure exactly I should implement this. How should I decide h?

[John D'Errico]

Why don't the derivest tools (the Hessian computation) work?

All it needs is that your function return a scalar, and that it be sufficiently smooth. If your function does not return a scalar, how do you define the Hessian matrix anyway? And if it is not smooth, how would you compute the derivatives you desire?

There is no need for your function to be in any symbolic form. A black box function is fine.

Answer 1

[Sean de Wolski]

Why not just let fmincon use finite differencing?

If that is not working, then it's possible that your function is discontinuous or non-differentiable and you should look at using one of the non gradient based solvers like patternsearch.

[xueqi]

Hi thank you. But in the documentation it says "You must supply the gradient of the objective function". I can not provide that as I can not get explicit expressions of the derivatives for my function.

I don't use patternsearch because it does not provide Hessian.

[Sean de Wolski]

No, if you want to use the analytical gradient, which you do not have, then you need to. Otherwise it will solve it with finite differences.

[Sean de Wolski]

x = fmincon(@sin,5,[],[],[],[], 0, 10)

No gradient or hessian supplied.