Wrong Hessian output in fminunc

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Alex
Alex 2020-8-12
评论： Alex 2020-8-17
Can somebody explain weird Matlab's output for the following optimization problem? The goal is to minimize and to evaluate Hessian matrix at the solution. However, Matlab returns a matrix of NaNs for the Hessian at the solution point. Here is the code
[xOpt,~,~,~,grad,hessian] = fminunc(@(x) x(1)^2+x(2)^2,[1,1])

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Matt J
Matt J 2020-8-13
编辑：Matt J 2020-8-13
Another solution would be to use standalone routines for numerical gradient and hessian estimation
https://www.mathworks.com/matlabcentral/fileexchange/13490-adaptive-robust-numerical-differentiation
fun=@(x) (x(1)^2+x(2)^2);
xOpt = fminunc(fun ,[1,1]);
g=gradest(fun,xOpt)
H=hessian(fun,xOpt)
No hidden scale factors if you go that route:
xOpt =
0 0
g =
0 0
H =
2.0000 0
0 2.0000
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Matt J
Matt J 2020-8-16
There is no requirement in the solution I've given you that the objective be in analytic form.
Alex
Alex 2020-8-17
Isn't is explicitly stated in the documentation for the files that you referenced?

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Bruno Luong
Bruno Luong 2020-8-12
编辑：Bruno Luong 2020-8-13
I guess the hessian is estimated from BFGS formula, that needs more than 1 FMINUNC iteration. In you case the minimum is reached in 1 iteration (since the gradient point directly to the minimum), the hessian estimation is not yet in the update cycle.
A small modification make # iteration > 1, and hessian starts well.
[xOpt,~,~,~,grad,hessian] = fminunc(@(x) 2*x(1)^2+x(2)^2,[1,1])
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Alex
Alex 2020-8-12
Thank you for the explanation. However, you modified the function to minimize and the Hessian is not the same as for the original problem. Do you know if there is any other way in Matlab to obtain a Hessian matrix at a point?
Bruno Luong
Bruno Luong 2020-8-13
编辑：Bruno Luong 2020-8-13
Not 100% satisfied answer but if you use trust-region algorithm and suppy the gradient, you 'll get back the hessian even with 1 iteration
options = optimoptions('fminunc', ...
'Algorithm', 'trust-region', ...
'SpecifyObjectiveGradient', true ...
);
[xOpt,out,b,c,grad,hessian] = fminunc(@fun,[1,1],options)
function [f,g] = fun(x)
% Calculate objective f
f = x(1)^2+x(2)^2;
if nargout > 1 % gradient required
g = 2*x;
end
end
If your function is quadratic and the -gradient points toward the minimum at any point, meaning if FMINUNC converge in 1 iteration regardless the starting point, then you must have hessian that is a scale of eye matrix. The NAN likely happens only for the trivial case as you have showed.

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Matt J
Matt J 2020-8-13
编辑：Matt J 2020-8-13
Just run fmincon (twice) with no constraints. Running a second time is important, because the hessian output is not the Hessian calculated at the final point, but rather the Hessian at the point just prior to that.
fun=@(x) x(1)^2+x(2)^2;
xOpt = fmincon(fun ,[1,1]);
[xOpt,~,~,~,~,grad,hessian] = fmincon(fun,xOpt)
xOpt =
1.0e-08 *
0.5588 0.5588
grad =
1.0e-07 *
0.2608
0.2608
hessian =
1 0
0 1
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Matt J
Matt J 2020-8-17
Alex's comment moved here:
Isn’t it explicitly stated in the documentation?
Matt J
Matt J 2020-8-17
Not as far as I can see. The documentation on the meaning of the Hessian output is here,
https://www.mathworks.com/help/optim/ug/hessian.html
but there is no mention that I can see of any pre-scaling of the objective.

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