Consider the following optimisation problem.
Given a set $S$ with $q$ weight functions $w_1, \ldots, w_q: S\rightarrow \mathbb{R}_+$ and a constant $1\leq k\leq |S|-1$. Find an $X\subset S, |X|=k$ subset of $S$ maximizing
$$ \min_{i=1\dots q} w_i(X) $$
$q=1$ and $|S|-1$ are trivial, and I am particularly interested in case $q=2$. Is it solvable in an elegant and fast way?
You can solve the problem via mixed integer linear programming as follows. For $s\in S$, let binary decision variable $x_s$ indicate whether $s\in X$. The problem is to maximize $z$ subject to linear constraints \begin{align} \sum_{s\in S} x_s &= k \tag1\label1 \\ z &\le \sum_{s\in S} w_i(s) x_s &&\text{for $i=1\dots,q$} \tag2\label2 \end{align} Constraint \eqref{1} selects $k$ elements. Constraint \eqref{2} linearizes the maximin objective.
