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.