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I'm trying to find the general way of optimizing a summation function and I'm a bit lost. I would like to find the maxium (and the minimum, ideally) of:

$$ \sum_{n=1}^{x} f(n)$$

For example, let's say:

$$ f(n) = -(n-4)^2 + 16 $$

In this scenario, the answer is easy to find without doing any work: f(n) is positive for $n \in [0, 8]$, and then becomes negative for infinity, we can maximize the sum by adding all positive numbers ($x = 8$).

Is there a way to find this mathematically, in a general way for all functions? Usually when looking to optimize a function I would take the derivative, but the derivative of a summation doesn't make much sense, does it?

Thanks.

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    Of course you can derive the summation, as long as it is finite. – 2017-02-07
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    Can you please remove the "accepted" tick from my answer? It isn't right, works by chance. :-) – 2017-02-07

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For a local max at positive integer $x$, you want $f(x) \ge 0$ while $f(x+1) \le 0$. Similarly for a local min with $\le$ and $\ge$. So you look for the places where $f$ changes sign.

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    Say for any $\epsilon > 0$ we can easily determine a sufficiently large $K(\epsilon) \in \mathbb{N}$ s.t. $\left| \sum_{n = K(\epsilon)}^\infty f(n)\right| < \epsilon,$ do you have any ideas on methods for efficiently finding a solution which is within $\epsilon$ of global optimality? – 2017-02-07
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    In general (i.e. if all you can do is evaluate $f$ at given values), then no: the best you can do is a brute-force search of $n = 0$ to $K(\epsilon)$. – 2017-02-08
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Attention, please: this solution works by chance. It isn't by any means right. Sorry.

When your summation is finite, you can take its derivative, and $$ \left(\sum_{n=0}^x f(n)\right)^\prime = \sum_{n=0}^x f^\prime(n). $$

Assuming $f(n) = -(n-4)^2 + 16$, we obtain $f^\prime(n) = -2n + 8$ and $$ \left(\sum_{n=0}^x f(n)\right)^\prime = \sum_{n=0}^x f^\prime(n) = 8x - 2\sum_{n=0}^x n, $$ which equals to $8x - x^2 - x = -x^2 + 7x$.

Solving for $x$, we find that $$ \begin{aligned} x(7-x) = 0 &\implies x = 0 \lor x = 7 \\ x(7-x) \ge 0 &\implies x \in [0,8). \end{aligned} $$

Why did $7$ come out, instead of $8$? My guess: for $n = 8$, $f(n)$ equals zero, meaning that the $8$-th term of the summation does not provide any contribute, so the local maximum is at $n = 7$.

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    I'm confused. He is trying to maximize over $x$ yet you seem to be differentiating w.r.t. $n$? Am I missing something obvious? I believe the OP was referring to $$\frac{d}{dx}\sum_{n=0}^x f(n)$$ in the question (which would, of course, be $f(x)$ under suitable conditions if we extended the summation to an integral). – 2017-02-07
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    To me, it seemed quite natural to derive first, and sum then. Doing the opposite brings almost the same result ($-x^2 + 7x -23/6$). To be honest, the more I ruminate on this, the more I get confused, too. – 2017-02-07
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    The problem is your differentiation is with respect to the wrong variable -- not that you are differentiating in the wrong order (differentiation is a linear operator). Your method worked by coincidence, it does not work in general. Consider $f(n) = (n-6)(n+4)$, then the summation is minimized at $x \in \{5,6\}$. However, note $$\frac{d}{dn}f(n) = 2n-2,$$ and $$\sum_{n=0}^x 2n-2$$ is monotone non-decreasing over the positive integers. – 2017-02-07
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    Ok. Thank you for the provided example, it made me understand what you meant in the previous comment. I'll leave the answer because I think it's useful to help others avoid this kind of mistake, and I'll request the OP to remove the green tick from this. – 2017-02-07