Fundamentals in Discrete Convex Analysis

This talk describes fundamental properties of M-convex and L-convex functions that play the central roles in discrete convex analysis. These concepts were originally introduced in combinatorial optimization, but turned out to be relevant in economics. Emphasis is put on discrete duality and conjugacy respect to the Legendre-Fenchel transformation. Monograph information: http://www.misojiro.t.u-tokyo.ac.jp/~murota/mybooks.html#DCAsiam200

On Equivalence of M♮^\natural-concavity of a Set Function and Submodularity of Its Conjugate

A fundamental theorem in discrete convex analysis states that a set function is M$^\natural$-concave if and only if its conjugate function is submodular. This paper gives a new proof to this fact

Discrete Convex Functions on Graphs and Their Algorithmic Applications

The present article is an exposition of a theory of discrete convex functions on certain graph structures, developed by the author in recent years. This theory is a spin-off of discrete convex analysis by Murota, and is motivated by combinatorial dualities in multiflow problems and the complexity classification of facility location problems on graphs. We outline the theory and algorithmic applications in combinatorial optimization problems

Time bounds for iterative auctions : a unified approach by discrete convex analysis

We investigate an auction model where there are many different goods, each good has multiple units and bidders have gross substitutes valuations over the goods. We analyze the number of iterations in iterative auction algo- rithms for the model based on the theory of discrete convex analysis. By making use of L♮-convexity of the Lyapunov function we derive exact bounds on the number of iterations in terms of the ℓ1-distance between the initial price vector and the found equilibrium. Our results extend and unify the price adjustment algorithms for the multi-unit auction model and for the unit-demand auction model, offering computational complexity results for these algorithms, and reinforcing the connection between auction theory and discrete convex analysis