M-Convexity and Its Applications in Operations

Ma-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 Ma-convexity and its variant SSQMa-convexity (semistrictly quasi Ma-convexity). We show that in a parametric maximization model, the optimal solution is nonincreasing in the parameters when the objective function is SSQMa-concave and the constraint is a box and illustrate when SSQMa-convexity and Ma-convexity are preserved. A sufficient and necessary characterization of twice continuously differentiable Ma-convex functions is provided. We then use them to analyze two important operations models: a classical multiproduct 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 Ma-convexity allows to simplify the complicated analysis in the literature for each model and extend the results to more general settings.

Automated summary

The paper discusses the properties of Ma-convexity and its variant SSQMa-convexity, as well as the nonincreasing nature of optimal solutions in certain parametric maximization models. It also provides a characterization of twice continuously differentiable Ma-convex functions and analyzes two important operations models.