non-linear optimization problem with constraints

Question

I have a trouble solving this non-linear optimization... it is difficult for me since it has two sets of variables $$ \text{i.e. } x_k \text{ and } y_k $$

\begin{align} f(X,Y)=\max_{x_k, y_k} \sum_{k=1}^L x_k\left(\frac{cy_k}{1+cy_k}\right) \\s.t. \sum_{k=1}^L x_k=La, \\ \sum_{k=1}^L y_k=Lb \\ a>0, b>0, c>0, x_k>0 \text{ and } y_k > 0 \end{align} and $a, b, c$ are constant value.

I know that \begin{align} f(Y)=\max_{y_k} \sum_{k=1}^L \left(\frac{cy_k}{1+cy_k}\right) \end{align} is concave function and has maximum when yk has same value i.e. $y_k=b$

here, can we say the function $f(X,Y)$ is

1. concave? and how can I determine $f(X,Y)$ is concave or convex?
2. maximized when $x_k$ and $y_k$ have equal value. i.e. $x_k=a, y_k=b$?

please, help me and I would really appreciate for your help

thanks

Answer 1

Here is one approach:

The keys here are linearity, convexity and monotonicity.

By scaling $x,y,c$ appropriately we can assume that $La = Lb = 1$.

Let $\Sigma = \{x | \sum_k x_k = 1, x_k \ge 0 \}$, $f(y) = {cy \over 1+cy}$. Note that $f$ is increasing on $[0, \infty)$.

The problem (I have relaxed the problem slightly) becomes \begin{eqnarray} \max_{y , x \in \Sigma} \sum_k x_k f(y_k) &=& \max_{y \in \Sigma} \max_{x \in \Sigma} \sum_k x_k f(y_k) \\ &=& \max_{y \in \Sigma} \max_k f(y_k) \\ &=& \max_k \max_{y \in \Sigma} f(y_k) \\ &=& \max_k f(1) \\ &=& f(1) \\ &=& {c \over 1+c} \end{eqnarray}