# Two-dimensional optimization: need an appropriate measure of confidence

I have a fitness function $f(x,y)$, which I am sampling at discrete $x$ and $y$ in order to minimize $f$. In addition to the location of the minimum, I would like to give an appropriate measure of confidence or quality of the determined location, so that I can detect if the minimum is rather wide, or if $f$ is non-convex.

Without making further assumptions about $f$, what would be an appropriate measure of quality?

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What method are you using to find the minimum? Or are you just sampling the function at many points and finding the minimum of this list? –  icurays1 Jan 25 '13 at 15:17
Indeed I'm sampling the function. –  Hendrik Jan 25 '13 at 15:19
I should add I'm sampling the function in a rectangular grid of $(x,y)$ values. –  Hendrik Jan 25 '13 at 15:20
Without further assumptions about $f$, you can say very little. It might be a rapidly oscillating function that you have undersampled, like $\sin(100x)$. –  Rahul Jan 27 '13 at 1:02
Okay, what if we exclude undersampling and assume that the function is steady? –  Hendrik Jan 27 '13 at 10:32