M♮-Convexity and Its Applications in Operations

A new approach for structural analysis of operations models with substitutability structures. In many operations models with substitutability structures, one often ends up with parametric optimization models that maximize submodular objective functions, and it is desirable to derive structural properties including monotone comparative statics of the optimal solutions or preservation of submodularity under the optimization operations. Yet, this task is challenging because the classical and commonly used results in lattice programming, applicable to optimization models with supermodular objective function maximization, do not apply. Using a key concept in discrete convex analysis, M♮-convexity, Chen and Li establish conditions under which the optimal solutions are nonincreasing in the parameters and the preservation property holds for parametric maximization models with submodular objectives, together with the development of several new fundamental properties of M♮-convexity. Their approach is powerful as demonstrated by applications in a classical multiproduct stochastic inventory model and a portfolio contract model.

On optimal taxes and subsidies: A discrete saddle-point theorem with application to job matching under constraints

When a government intervenes in markets by setting a target amount of goods/services traded, its tax/subsidy policy is optimal if it entices the market participants to obey the policy target while achieving the highest possible social welfare. For the model of job market interventions by Kojima et al. (2019), we establish the existence of optimal taxes/subsidies as well as their characterization. Our methodological contribution is to introduce a discrete version of Karush-Kuhn-Tucker's saddle-point theorem based on the techniques in discrete convex analysis. We have two main results: we (i) characterize the optimal taxes/subsidies and the corresponding equilibrium salaries as the minimizers of a Lagrange function, and (ii) prove that the function satisfies a notion of discrete convexity (called L#-convexity). These results together with others imply that an optimal tax/subsidy level exists and can be calculated via a computationally efficient algorithm.

On the Construction of Substitutes

Gross substitutability is a central concept in economics and is connected to important notions in discrete convex analysis, number theory, and the analysis of greedy algorithms in computer science. Many different characterizations are known for this class, but providing a constructive description remains a major open problem. The construction problem asks how to construct all gross substitutes from a class of simpler functions using a set of operations. Because gross substitutes are a natural generalization of matroids to real-valued functions, matroid rank functions form a desirable such class of simpler functions. Shioura proved that a rich class of gross substitutes can be expressed as sums of matroid rank functions, but it is open whether all gross substitutes can be constructed this way. Our main result is a negative answer showing that some gross substitutes cannot be expressed as positive linear combinations of matroid rank functions. En route, we provide necessary and sufficient conditions for the sum to preserve substitutability, uncover a new operation preserving substitutability, and fully describe all substitutes with at most four items.

Product-Mix Auctions and Tropical Geometry

In a recent and ongoing work, Baldwin and Klemperer explore a connection between tropical geometry and economics. They give a sufficient condition for the existence of competitive equilibrium in product-mix auctions of indivisible goods. This result, which we call the unimodularity theorem, can also be traced back to the work of Danilov, Koshevoy, and Murota in discrete convex analysis. We give a new proof of the unimodularity theorem via the classical unimodularity theorem in integer programming. We give a unified treatment of these results via tropical geometry and formulate a new sufficient condition for competitive equilibrium when there are only two types of products. Generalizations of our theorem in higher dimensions are equivalent to various forms of the Oda conjecture in algebraic geometry.

Uncovering the effective interval of resolution parameter across multiple community optimization measures

The study of community structure is a primary focus of network analysis, which has attracted a large amount of attention. In this paper, we focus on two famous functions, i.e., the Hamiltonian function [Formula: see text] and the modularity density measure [Formula: see text], and intend to uncover the effective thresholds of their corresponding resolution parameter [Formula: see text] without resolution limit problem. Two widely used example networks are employed, including the ring network of lumps as well as the ad hoc network. In these two networks, we use discrete convex analysis to study the interval of resolution parameter of [Formula: see text] and [Formula: see text] that will not cause the misidentification. By comparison, we find that in both examples, for Hamiltonian function [Formula: see text], the larger the value of resolution parameter [Formula: see text], the less resolution limit the network suffers; while for modularity density [Formula: see text], the less resolution limit the network suffers when we decrease the value of [Formula: see text]. Our framework is mathematically strict and efficient and can be applied in a lot of scientific fields.