Locally Farkas–Minkowski linear inequality systems

This paper introduces the locally Farkas–Minkowski (LFM) linear inequality systems in a finite dimensional Euclidean space. These systems are those ones that satisfy that any consequence of the system that is active at some solution point is also a consequence of some finite subsystem. This class includes the Farkas–Minkowski systems and verifies most of the properties that these systems possess. Moreover, it contains the locally polyhedral systems, which are the natural external representation of quasi-polyhedral sets. The LFM systems appear to be the natural external representation of closed convex sets. A characterization based on their properties under the union of systems is provided. In linear semi-infinite programming, the LFM property is the more general constraint qualification such that the Karush–Kuhn–Tucker condition characterizes the optimal points. Furthermore, the pair of Haar dual problems has no duality gap.