This is my first time asking an optimization question on here, so I am looking forward to see what will happen here.
In the lack of a better title, I wrote it as it is. At a high-level, I can perhaps describe the problem at hand as a two-level or two-stage problem. I will try to describe it very generally, to hear if there are any ideas immediately. Otherwise, I guess I will provide more detail.
Firstly, I have a number of constants: $R,C,T,\rho \in \mathbb{Z}_+$.
Assume there is a finite set of matrices $\Pi = \{\pi_1,\ldots,\pi_{|\Pi|}\}$ of size $\rho \times R$. The matrices $\pi_i \in \Pi$ will have a special form, and in particular the elements are from $\{0,\ldots,C-1\}$. For each $\pi_i \in \Pi$, we can consider $\pi_i$ as a (constant) parameter to a linear programming optimization problem ${\cal O}(\pi_i,R,C,\rho,T)$. In this optimization problem, we are interested in minimizing an associated value which we denote $z_i$.
What I am really interested in, is determining the $\pi_i \in \Pi$, for which the associated $z_i$ is as large as possible, i.e. $max_{\pi_i \in \Pi}\{z_i\}$, where $z_i$ themselves represent minimization LP problems.
So maybe it is clear why this can be seen as a two-level problem: On one level I want, given a fixed $\pi_i \in \Pi$, to determine the smallest possible $z_i$, and on the other level I want to find the $\pi_i \in \Pi$ which maximizes the $z_i$.
I have close to no idea where to start on this. I've formulated the associated LP problem, but where the $\pi_i$ is itself a decision variable. The way it is formulated, I can not simply throw it to CPLEX. One could consider simply solving the LP problem for each $\pi_i \in \Pi$, but in some instances $|\Pi|$ will be so large that this is infeasible.
Any ideas or suggestions on existing work is much appreciated!