1
$\begingroup$

Consider the following optimization problem:

$u^*(k) = \arg \max\limits_{0 \le u \le k} \; f(u, k) \; , \; u,k \in \mathbb{Z}^+$,

where, $f(u,k)$ has non-increasing increments in $k$, that is, $(f(u,k+1) - f(u, k))$ is non-increasing in $k$. Also, for each $u$ I am given that $f(u,k)$ is a concave function in $k$; furtheremore,

$f(u, k+1) \ge f(u, k)$.

I need to know what is the relation between $u^*(k+1)$ and $u^*(k)$? Intuitively, I feel like I can claim the following

(i) if $0 \le u^*(k+1) \le k$, then $u^*(k+1) = u^*(k)$,

(ii) if $u^*(k+1) = k+1$, then $u^*(k) = k$.

But I am not sure and haven't been able to establish this either. Any insights on this would be really appreciated.

  • 0
    There does not seem to be a restriction as to how $u$ influences $f$. For each $u$ you could have a completely different concave non-decreasing function in $k$. I therefore do not think there is any relation between $u*(k)$ for different $k$. – 2017-01-21

0 Answers 0