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I have the following problem:

A reward function $R(i)$ defined as

$$ R(i) > R(j) \iff i > j $$

and a cost function $C(i)$

$$ C(i) > C(j) \iff i > j $$

Both functions are defined on positive integers.

I cannot change the sequence $C(i)$ for $i \in 0,\ldots,n$ but I can choose any arbitrary reward function $R$. The only constraints is the increasing monotonicity.

Then I have to find the $i$ value that maximizes $R$, while minimizing $C$.

Or more formally the largest value of i that provided "bounded" increase in both cost and rewards. For example, if C and R are defined for $i \in 0,\ldots,5$ in this way

$$ R(i) = [0,1,2,3,4,5] $$

and $$ C(i) = [10,12,15,44,48]$$

then the best $i$ would be 2, as the increase in cost does not compensate the increase in reward.

Which tool I may use in this case?

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    You want to maximize $R(i)-C(i)$? – 2017-01-31

1 Answers 1

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You cannot maximize $R$ and minimize $C$ at the same time. The smallest value of $C(i)$ is defined on $i=0$, while the largest value of $R(j)$ appears at $j=n$.

Basically, you need to formalize your sentence

Then I have to find the $i$ value that maximizes $R$, while minimizing $C$.

because it is mathematically imprecise. There is only one value of $i$ which minimizes $C$, and that value is $i=0$ regardless of what $R$ and $C$ are. For values of $i\neq 0$, you have $C(i)>C(0)$, so those values don't minimize $C$.

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    I understand. What about maximizing the fairness between R and C? Something like each increase in $i$ should have a bounded delta in both R and C? I also added an example – 2017-01-31
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    @M.G. Your example is very vague, so still not formalized enough. I don't know what you mean by "does not compensate the increase in reward". Your question isn't *yet* a mathematics question, it's a question of definitions. – 2017-01-31