Optimizing the max-min function with a constraint on a two-sided linear system

The behavior of discrete-event systems, in which the individual components move from event to event rather than varying continuously through time, is often described by systems of linear equations in max-min algebra, in which classical addition and multiplication are replaced by $\oplus$ and $\otimes$, representing maximum and minimum, respectively. Max-min equations have found a broad area of applications in causal models, which emphasize relationships between input and output variables. Many practical situations can be described using max-min systems of linear equations. We shall deal with a two-sided max-min system of linear equations with unknown column vector $x$ of the form $A\otimes x\oplus c=B\otimes x\oplus d$, where $A$, $B$ are given square matrices, $c$, $d$ are column vectors and operations $\oplus$ and $\otimes$ are extended to matrices and vectors in the same way as in the classical algebra. We give an equivalent condition for its solvability. For a given max-min objective function $f$, we consider optimization problem of type $f^{\mathrm{\top }}\otimes x\to max\text{or}min$ constraint to $A\otimes x\oplus c=B\otimes x\oplus d$. We solve the equation in the form $f(x)=v$ on the set of solutions of the equation $A\otimes x\oplus c=B\otimes x\oplus d$ and extend the problem to the case of an interval function $\mathit{f}$ and an interval value $\mathit{v}$. We define several types of the reachability of the interval value $\mathit{v}$ by the interval function $\mathit{f}$ and provide equivalent conditions for them.