M-natural-Convexity and its Applications in Operations

Abstract

M-natural-convexity, one of the main concepts in discrete convex analysis, possesses many salient structural properties and allows for the design of efficient algorithms. In this paper, we establish several new fundamental properties of M-natural-convexity and its variant SSQM- natural-convexity (semistrictly quasi M-natural-convexity). We show that in a parametric maximization model, the optimal solution is nonincreasing in the parameters when the objective function is SSQM- natural-concave and the constraint is a box, and illustrate when SSQM- natural-convexity and M-natural-convexity are preserved. A sufficient and necessary characterization of twice continuously differentiable M- natural-convex function is provided. We then utilize them to analyze two important operations models: a classical multi-product dynamic stochastic inventory model, and a portfolio contract model where a buyer reserves capacities in blocks from multiple competing suppliers. We illustrate that looking from the lens of M-natural-convexity allows to simplify the complicated analysis in the literature for each model and extend the results to more general settings.