ALGORITHMS FOR L-CONVEX FUNCTION MINIMIZATION: CONNECTION BETWEEN DISCRETE CONVEX ANALYSIS AND OTHER RESEARCH FIELDS

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 present algorithms for L-convex function minimization. Algorithms proposed independently in research fields such as discrete optimization, auction theory, and computer vision can be regarded as minimization algorithms applied to specific L-convex functions. This fact indicates the close connection between discrete convex analysis and these research fields. We then theoretically analyze the number of iterations required by some minimization algorithms, and show that the precise bounds can be given in terms of distance between the initial solution and the minimizer found by the algorithms. This fact implies that the algorithms output the “nearest” minimizer to the initial solution, and that the trajectory of solutions generated by the algorithms are “shortest paths” from the initial solution to the found minimizer. In this way, we can provide a unified viewpoint to algorithms appearing in various research fields by analyzing steepest descent algorithms for an L-convex function. Finally, we apply the analysis results to iterative auctions in auction theory. We show that the essence of the iterative auctions proposed by Ausubel (2006) lies in L-convexity, and using this fact we analyze the behavior of the iterative auctions. We also review the iterative auctions proposed by Murota–Shioura–Yang (2016), which are based on the understanding of discrete convex analysis.