This technical note utilizes Minkowski algebra of convex sets to characterize a family of local control Lyapunov functions for constrained linear discrete-time systems. Local control Lyapunov functions are induced by parametrized contractive invariant sets. Underlying contractive invariant sets belong to a family of Minkowski decomposable convex sets and are, in fact, parametrized by a basic shape set and linear transformations of system matrices and a set of design matrices. Corresponding local control Lyapunov functions can be detected by solving a single, tractable, convex optimization problem which in case of polyhedral constraints reduces to a single linear program. The a priori complexity estimate of the characterized local control Lyapunov function is provided for some practically relevant cases. An illustrative example and relevant numerical experience are also reported.