2017 Volume 60 Issue 3 Pages 216-243
L-convexity is a concept of discrete convexity for functions defined on the integer lattice points, and plays a central role in the framework of discrete convex analysis. In this paper, we review recent development of algorithms for L-convex function minimization. We first point out the close connection between discrete convex analysis and various research fields such as discrete optimization, auction theory, and computer vision by showing that algorithms proposed independently in these research fields can be regarded as minimization algorithms applied to specific L-convex functions. Therefore, we can provide a unified approach to analyze the algorithms appearing in various research fields through the concept of L-convex function. We then present the recent results on theoretical bounds of the number of iterations required by some minimization algorithms, where precise bounds are given in terms of distance between the initial solution and the minimizer found by the algorithms. From these results we see that the algorithms output the “nearest” minimizer to the initial solution, and that the trajectories of solutions generated by the algorithms are “shortest paths” from the initial solution to the found minimizer. Finally, we consider an application of the results to iterative auctions in auction theory. We point out that the essence of the iterative auctions proposed by Ausubel (2006) lies in L-convexity. We also present new iterative auctions by Murota–Shioura–Yang (2016), which are based on the understanding of existing iterative auctions from the viewpoint of discrete convex analysis.
