Proximity theorems of discrete convex functions

A proximity theorem is a statement that, given an optimization problem and its relaxation, an optimal solution to the original problem exists in a certain neighborhood of a solution to the relaxation. Proximity theorems have been used successfully, for example, in designing efficient algorithms for discrete resource allocation problems. After reviewing the recent results for L-convex and M-convex functions, this paper establishes proximity theorems for larger classes of discrete convex functions, L2-convex functions and M2-convex functions, that are relevant to the polymatroid intersection problem and the submodular flow problem.