Discrete L-/ M-Convex Function Minimization Based on Continuous Relaxation

We consider the problem of minimizing a nonlinear discrete function with L-/M-convexity proposed in the theory of discrete convex analysis. For this problem, steepest descent algorithms and steepest descent scaling algorithms are known. In this paper, we use continuous relaxation approach which minimizes the continuous variable version first in order to find a good initial solution of a steepest descent algorithm. For discrete L-/M-convex functions, we give proximity theorems showing that a discrete global minimizer exists in the neighborhood of a continuous global minimizer. These proximity theorems afford theoretical guarantees for the efficiency of the proposed algorithms.