I have a minimax problem of the form $$\max_{\substack{u_1,\dots,u_n \ge 0 \\ u_1+\dots+u_n = 1}} \min_{\substack{v_1,\dots,v_m \ge 0 \\ v_1+\dots+v_m = 1 \\ v_{j_1} \le v_{j_2} \hspace{1mm} \forall (j_1,j_2) \in J}} \sum_{i=1}^n \sum_{j=1}^m c_{i,j} u_i v_j,$$ where $J$ is some (given) subset of $[m]\times[m]$ and each $c_{i,j} \in \{0,1\}$ (is given).
So it's a normal kind of minimax problem, except we have some inequalities amongst the $v_j$'s.
By the minimax theorem, there are $u^* = (u_i^*)_{i=1}^n, v^* = (v_j^*)_{j=1}^m$ that are ``best responses" to one another.
Let's assume our problem is such that there is only one such $u^*$.
Question: How can I quickly (in practice, on a computer) find $u^*$?
The values of $n$ and $m$ I am working with are actually very small (e.g. $n \le 7, m \le 14$). However, Mathematica seemingly takes forever to compute the minimax, even with $n=2,m=4$.
Using Mathematica, I can find (some) $v^*$ rather quickly by interchanging the max and the min, and then noting that the max of the double sum over the possible $(u_i)_{i=1}^n$'s is simply a max of the double sum over the $n$ tuples $(u_i)_{i=1}^n$ with exactly one $1$ (and the rest $0$). However, this doesn't help me find $u^*$.
I'd appreciate any advice. Thanks!
