Technical efficiency and distance to a reverse convex set

The problem of finding the distance to a reverse (or complement of a) convex subset in a normed vector space is considered. This nonconvex and, in general, nonsmooth optimization problem arises in quantitative economics in the theory of measuring the technical efficiency of production units. In this context, applying a suitable duality theorem similar to the Nirenberg's one known for the distance to a convex subset, the problem reduces to a finite number of independent linear programming problems.