# Why fminunc does not find the true global minimum?

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MRC on 3 Feb 2014
Commented: MRC on 3 Feb 2014
Hi all, I should solve this unconstrained optimization problem (attached). I know that the function has the global minimum at [1 2 2 3]. However, if I set as starting value [1 2 2 3], the algorithm ends up at [1.1667 2.4221 2.2561 3]. I have some doubts to clarify (I'm not familiar with this topic, sorry for my trivial questions):
1) The algorithm output reveals that at iteration 0 the function takes value 5.47709e-06 and at iteration 10 the function takes value 1.41453e-06. But, if I compute the function value at [1 2 2 3] I get 1.4140e-06 and if I compute the function value at [1.1667 2.4221 2.2561 3] I get 1.5635e-06. Why are these values different from the starting and final function values reported in the algorithm output?
2) How can I force the algorithm to keep searching until it arrives at [1 2 2 3]?
Thanks!

Matt J on 3 Feb 2014
When you call fminunc with all 5 outputs
what are the values of these outputs?
In particular, if the true Hessian is singular at the global min [1 2 2 3], I can imagine its finite difference estimate, as computed by fminunc, could be numerically problematic, e.g., not positive definite.
MRC on 3 Feb 2014
thank you!

Alan Weiss on 3 Feb 2014
I did not look at your data. But I doubt that the true global minimum is at [1 2 2 3] if you are really fitting to data. I would bet that you generated data from a known distribution, and then fit the model to that data. You will never get perfect match to the initial distribution, because the data that you used is not perfectly distributed according to the theoretical distribution.
For instance, this toolbox example shows theoretical parameters of [1 3 2], and yet the fitted model has parameters [1.0169 3.1444 2.1596], and the fitted model is at a global minimum for that data set.
Alan Weiss
MATLAB mathematical toolbox documentation
MRC on 3 Feb 2014
If the function I maximized was the sample log-likelihood function for Y|X~f(theta), then you were right; but the function that I maximize is the expected log-likelihood and the data are just the X I'm conditioning on. I'm sure that the expected log-likelihood is maximized at [1 2 2 3] (I have some theoretical results which actually show this). Is there anyone who can help me in answering questions 1 and 2?