Let $r$ and $N$ be positive integers, and let $\epsilon$ be a positive real number for which $2\leq r \leq N-1$ and $\frac{\epsilon}{N-r}\leq \frac{1-\epsilon}{r}$. I would like to analytically solve the following univariate minimization problem:
Minimize $$ f(p)=\binom{r+k-1}{r+1}p^{r+1} + \binom{r+k-1}{r}\big[1-(r+k-1)p\big]p^r+\binom{r+k-1}{r-1}\big[1-\epsilon-(1-r-1)p\big]\big[\epsilon-pk\big]p^{r-1} $$ Subject to $\frac{\epsilon}{N-r}\leq p \leq \frac{1-\epsilon}{r}$ and $k=\lfloor\epsilon/p\rfloor.$
This seems tough, given that $k$ is a step function of $p$. My plan of attack is to break the problem into two steps:
- Prove that for any fixed positive integer $\frac{r\epsilon}{1-\epsilon}\leq k\leq N-r$, it holds that
$$ \min_{\frac{\epsilon}{k-1}\leq p\leq \frac{\epsilon}{k}} f(p)=\min\big\{f\left(\frac{\epsilon}{k-1}\right),f\left(\frac{\epsilon}{k}\right)\big\}, $$ i.e. that the minimum is attained for $p$ such that $\epsilon/p$ is an integer.
- Minimize $$ g(k):=\binom{r+k-1}{r+1}\left(\frac{\epsilon}{k}\right)^{r+1} + \binom{r+k-1}{r}\big[1-(r+k-1)\frac{\epsilon}{k}\big]\left(\frac{\epsilon}{k}\right)^r $$ over all integers $\frac{r\epsilon}{1-\epsilon}\leq k\leq N-r$.
I have verified step 1, but I don't know how to approach step 2.
Any ideas or solutions would be appreciated!