  Differential positivity and K-cooperativity, a special case of differential
positivity, extend differential approaches to control to nonlinear systems with
multiple equilibria, such as switches or multi-agent consensus. To apply this
theory, we reframe conditions for strict K-cooperativity as an optimization
problem. Geometrically, the conditions correspond to finding a cone that a set
of linear operators leave invariant. Even though solving the optimization
problem is hard, we combine the optimization perspective with the geometric
intuition to construct a heuristic cone-finding algorithm centered around
Linear Programming (LP). The algorithm we obtain is unique in that it modifies
existing rays of a candidate cone instead of adding new ones. This enables us
to also take a first step in tackling the synthesis problem for K-cooperative
systems. We demonstrate our approach on some examples, including one in which
we repurpose our algorithm to obtain a novel alternative tool for computing
polyhedral Lyapunov functions of bounded complexity.
