I have the following optimisation question. I was wondering if you could help me solve it.
There are $K$ different sets of data items. Set $k$ for example, has $N_k$ data items. The request rate for data items in set $k$ follows a Poisson distribution with $\lambda_k = \sum_{i=1}^{N_k}\lambda_{ik}$ in which $\lambda_{ik}$ is the request rate of item $i$ in set $k$ .
A cache with size of $C_k$ data items is dedicated to set $k$. The hit ratio of for item $i$ in set $k$ for cache of size $C_k$ could be calculated as follows: $h_{ik} = (1-e^{-\lambda_{ik}\tau_{k}})$;
in which $\tau_k$ could be obtained as follows:
$C_k = \sum_{i=1}^{N_k}(1-e^{-\lambda_{ik}\tau_{k}})$. The overall hit rate for cache $k$ then could be calculated through $h_k = \sum_{i=1}^{N_k} \lambda_{ik}(1-e^{-\lambda_{ik}\tau_k})$. It is obvious that $h_k$ is a function of $C_k$; so we show $h_k$ with $h_k(C_k)$.
Associated with each set $k$ is a utility function $U_k(h_k)$. Each $U_k(·)$ is assumed to be a continuous, monotonically increasing, and concave function.
Could someone please help me with a suggestion or hint to help me solve the following optimisation question?:
\begin{equation} \underset{C_1,\dots,C_K}{\text{maximize}} \sum_{k=1}^{K}U_k\big(h_k(C_k)\big) \end{equation}
\begin{equation} \text{subject to:} \sum_{k=1}^K C_k= C \end{equation}
Thanks in advance.