I want to solve the following optimization problem. Suppose we are given $p_r^i \in [0,1]$ for $r={1,2,...,N}$ and $i={1,2}$ such that $\sum_{r=1}^N p_r^i =1$ for i={1,2}. We want to find $x_r \in [0,1]$ and $y_r \in [0,1]$, $r={1,2,...,N}$ that maximize
$\sum_{k=1}^N \sum_{l=1}^N (x_k + y_l + min\{ 1-x_k, y_k,1-y_l,x_l\})p_k^1 p_l^2$
subject to $\sum_{r=1}^N x_r =1$,$\sum_{r=1}^N y_r =1$.
Possibly we can rewrite the problem as follows: find $x_r \in [0,1]$ and $y_r \in [0,1]$, $r={1,2,...,N}$ that maximize
$\sum_{k=1}^N \sum_{l=1}^N (x_k + y_l + z_{kl})p_k^1 p_l^2$
subject to
$\sum_{r=1}^N x_r =1$,$\sum_{r=1}^N y_r =1$.
$z_{kl} \leq 1-x_k$ for $k =1,2,...,N$
$z_{kl} \leq y_k$ for $k =1,2,...,N$
$z_{kl} \leq 1-y_l$ for $l =1,2,...,N$
$z_{kl} \leq 1-x_l$ for $l =1,2,...,N$
Any idea for finding optimal solution as function of the parameters involved in the problem?