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I have this equation:

$f(k)=argmax_{0\le T\le T_{max}} \left(X+\left\lceil\frac{k-X}{T}\right\rceil T\right)$

$X$ is uniformly distributed between $(0,T)$. I guess this is called a stochastic optimisation problem, but I am unable to figure out which method to use to solve this problem.

Is there a way to characterise the random variable $f(k)$? I am not sure if I could substitute $\frac{T}{2}$ in place of $X$ and solve the problem.

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    Yes, it will. If $T_{max}$ is larger than $1$, the maximum will be in the interval $[0,1]$ I guess, but the supremum will be in $[1,T_{max}]$. Just plot the part with the ceiling functions for some values of $k$, $X$ and $T_{max}$.2012-06-01

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