Calculate minimum distance between point and steady but NON differentiable function

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Dario Walter 2019-7-12

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评论： infinity 2019-7-15

Hey,

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?

Thanks for your help!

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Answer 1

infinity 2019-7-12

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Hello,

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.

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Dario Walter 2019-7-15

编辑：Dario Walter 2019-7-15

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 2019-7-15

Hello,

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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