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I have an array of length M. The elements of the array are either zero or positive real numbers. I need to derive a function/analytic expression (preferably linear/convex/concave) that finds the index of the first positive element of the array.

Example: Let p denote the desired array and let p = [0 0 0.3 2.6 0 0 1.1 0 1.8 0]

Here, the first positive element is 0.3 and its index is 3. I need an analytic expression that gives me this index, i.e., 3.

Previous work: I have already derived an expression for arrays that contain only binary numbers.

Let x denote the desired vector and x = [0 0 1 1 0 0 1 0 1 0].

let M denote the set of indices of x, i.e., M = {1,2,3,...,10}. Assume that x_m denotes the m-th entry of the array x. For example, x_2 = 0, x_3 = 1. Here, |M| = 10. The following expression (written in Latex format) finds the index of the first positive element of this array.

min_{m \in M} (m * x_m + |M| (1 - x_m))

= min (1 * x_1 + 10 * (1 - x_1), 2 * x_2 + 10 * (1 - x_2), ...)

= 3.

This function is concave and it provides the correct answer. Unfortunately, it only works for arrays that contain binary numbers!

I want to find an expression that works for arrays with real numbers (like array p). This function will become a constraint in my optimization problem. Hence, convexity/concavity is desired. However, any analytic expression that gives the index of the first positive element will help.

Any feedback will be appreciated.

Thanks,

Nazmul

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It seems the following.

You can simply define the expression for the required function $f$. For this simply replace each “x_i” in the function for binary array by “sign x_i”. From the other side, such the function $f$ is unique, and therefore its linearity or convexity does not depend of the form of its analytic expression, and you can easily check that the function $f$ is convex, because for any arrays $x,y$ such that $f(x)\le f(y)$ and $\lambda\in (0,1]$ we have that $f(\lambda x+(1-\lambda)y)=f(x)\le \lambda f(x)+(1-\lambda)f(y)$.

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