# Maximization over minimum function

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?

• Is there a particular context behind this problem? You can always solve it with integer programming (after linearization of the problem). – Kuifje Feb 10 '16 at 3:53
• I didn't get your meaning by particular context. About the solution approach you suggested, can you explain more? – m0_as Feb 10 '16 at 3:57
• What I meant was where does this problem come from? Are you familiar with integer programming? – Kuifje Feb 10 '16 at 4:04
• I just modeled a part of a problem on which I am working as this optimization problem. Regarding inter programming, I somewhat familiar, but not that much. So I think studying integer programming should be starting point. Thanks – m0_as Feb 10 '16 at 4:27
• @Kuifje As you suggested, I edited my question. Any idea how I can find the solution? – m0_as Feb 10 '16 at 7:27