Let a system S=(G=(V,E),w,F) consist of a digraph G
(not necessarily acyclic) with a set V of vertices and a set E of edges,
a weight function w from the union V and E to the set of reals
and a set F of functions fv from the set of reals to the set of reals for
each vertex v in V,
where w(u,v) denotes the weight of an edge (u,v)
from a vertex u in V and a vertex v in V.
A solution to system S is defined to be a set of reals y_v, v in V
such that y_v= fv(x), where x is w(v) plus +w(u,v)y_u over all edges (u,v) in E.
Finding solutions to a given system has an important application
in Artificial Neural Network (ANN).
In this paper, we show that when each function fv is a
continuous piece-wise linear function,
the problem of finding a solution to a system S
can be formulated as a Mixed Integer Linear Programming Problem (MILP)
with O(|V| +n') variables and constraints,
where n' denotes the total number of break points
over all functions fv, v in V.
Based on this, we can solve the inverse problem to an ANN N
as an MILP after approximating the activation function in N
as a piece-wise linear function.