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Calculate minimum distance between point and steady but NON differentiable function

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Dario Walter
Dario Walter on 12 Jul 2019
Commented: infinity on 15 Jul 2019
I could not find any mathematical explanation regarding this topic on the internet and hope for your help. Is there a way to determine the minimum between a point P and a piecewise function, e.g.
syms x
A = piecewise(964<=x<=5007,((2450-10)/(5007-964))*(x-964)+10,x>5007,2450);
P = [4900;3000]
For a straight line this is normally done by finding the vertical to the envelope. But how does it look for a NON differentiable function? Is there only an iterative solution?Mode1_Outer.jpgThanks for your help!


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Accepted Answer

infinity on 12 Jul 2019
There is two options that you can think
Assume we have a point P(xp, yp) and a function f(x) that we may not have derivative. Then,
1. we can define a distant between P and f by
Now, what we have to do is to find the minimum of d(x).
There are several approaches in Matlab can help you find the minimum of d.
2. We can split piecewise function into several intervals such that in each interval the function has derivative and we can find distance from P to each line and also at the intersection. Then, we can compare these solution to find the global minimum.


Dario Walter
Dario Walter on 15 Jul 2019
Thank you Trung.
Regarding 1. Usually you find the minimum by calculating the derivative. However, this is not possible in the present case.
2. This might work
infinity on 15 Jul 2019
If you might want to use gradient based method to find the minimum. The method 1 should not work. However, you could use direct optimization (non - based gradient) to find the minimum of a given function without care about smooth property. For example, genetic algorihtm which you could find in matlab. For this small problem, you might use "fminbnd" in matalb.
Hope it could help you.

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