ON POLYHEDRAL APPROXIMATION OF L-CONVEX AND M-CONVEX FUNCTIONS

Abstract

In discrete convex analysis, L-convexity and M-convexity are defined for functions in both discrete and continuous variables. Polyhedral L-/M-convex functions connect discrete and continuous versions. Specifically, polyhedral L-/M-convex functions with certain integrality can be identified with discrete versions. Here we show another role of polyhedral L-/M-convex functions: every closed L-/M-convex function in continuous variables can be approximated by polyhedral L-/M-convex functions, uniformly on every compact set. The proof relies on L-M conjugacy under the Legendre-Fenchel transformation.