# fminunc initial point is local minimum, but fminsearch returns reasonable estaimtes

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Tian on 15 Sep 2021
Commented: Tian on 15 Sep 2021
Hi,
I have a dataset for individual i and time t. I try to find 3 parameters w,a,b to minimize an objective function. Given the parameter values and data, the objective function first compute an optimal decision , then compute the sum of squared difference between and the observed choice X.
That is, I try to do: When I use fminsearch, it does return reasonable estiamtes around different starting values. The final points also have lower objective values.
However, fminunc always say "Initial point is a local minimum" and the Hessian is all 0s. I've tried (1) other starting values, (2) change the optimality tolerance to 10e-12, but the first-order optimality is 0 at starting values.
Since fminsearch does go to other points with lower objectives, does this mean my objective function isn't actually flat but fminunc just doesn't work well?
I do want to use fminunc to get the Hessian matrix... How can I debug/fix this?

Matt J on 15 Sep 2021
Edited: Matt J on 15 Sep 2021
It could happen if your objective function is piece-wise flat (and hence non-differentiable). fminsearch is a derivative-free solver, so it is less vulnerable to this, but a piece-wise flat objective is best avoided.
t=linspace(-5,5,1e6);
fun=@(x) interp1(t,t.^2,x,'nearest'); %piece-wise constant
tmin=fminunc(fun,3.5)
Initial point is a local minimum. Optimization completed because the size of the gradient at the initial point is less than the value of the optimality tolerance.
tmin = 3.5000
tmin=fminsearch(fun,3.5)
tmin = -1.3323e-15
However, now let's make the objective smooth:
fun=@(x) interp1(t,t.^2,x,'cubic'); %smoothed version of previous objective
tmin=fminunc(fun,3.5)
Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance.
tmin = -8.2595e-08
tmin=fminsearch(fun,3.5)
tmin = -1.3323e-15
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Tian on 15 Sep 2021
I see. Thank you for the detailed example!