M-Convex Function Minimization by Continuous Relaxation Approach: Proximity Theorem and Algorithm

The concept of M-convexity for functions in integer variables, introduced by Murota (1995), plays a primary role in the theory of discrete convex analysis. In this paper, we consider the problem of minimizing an M-convex function, which is a natural generalization of the separable convex resource allocation problem under a submodular constraint and contains some classes of nonseparable convex function minimization on integer lattice points. We propose a new approach for M-convex function minimization based on continuous relaxation. We show proximity theorems for M-convex function minimization and its continuous relaxation, and develop a new algorithm based on continuous relaxation by using the proximity theorems. The practical performance of the proposed algorithm is evaluated by computational experiments.