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Directed discrete midpoint convexity

JAPAN JOURNAL OF INDUSTRIAL AND APPLIED MATHEMATICS 2021年第1期38卷 1-37页

作者： Tamura, Akihisa Tsurumi, Kazuya Keio Univ Dept Math Yokohama Kanagawa 2238522 Japan

For continuous functions, midpoint convexity characterizes convex functions. By considering discrete versions of midpoint convexity, several types of discrete convexities of functions, including integral convexity, L.-convexity and global/local discrete midpoint convexity, have been studied. We propose a new type of discrete midpoint convexity that lies between L.-convexity and integral convexity and is independent of global/local discrete midpoint convexity. The new convexity, named DDM-convexity, has nice properties satisfied by L.-convexity and global/local discrete midpoint convexity. DDM-convex functions are stable under scaling, satisfy the so-called parallelogram inequality and a proximity theorem with the same small proximity bound as that for L.-convex functions. Several characterizations of DDMconvexity are given and algorithms for DDM-convex function minimization are developed. We also propose DDM-convexity in continuous variables and give proximity theorems on these functions.

关键词： Midpoint convexity Discrete midpoint convexity Integral convexity L-(sic)-convexity proximity theorem Scaling algorithm

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Discrete Midpoint Convexity

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MATHEMATICS OF OPERATIONS RESEARCH 2020年第1期45卷 99-128页

作者： Moriguchi, Satoko Murota, Kazuo Tamura, Akihisa Tardella, Fablo Tokyo Metropolitan Univ Dept Econ & Business Adm Tokyo 1920397 Japan Keio Univ Dept Math Yokohama Kanagawa 2238522 Japan Sapienza Univ Rome Dept Methods & Models Econ Terr & Finance I-00161 Rome Italy

For a function defined on the integer lattice, we consider discrete versions of midpoint convexity, which offer a unifying framework for discrete convexity of functions, including integral convexity, L-(sic)-convexity, and submodularity. By considering discrete midpoint convexity for all pairs at l(infinity)-distance equal to 2 or not smaller than 2, we identify new classes of discrete convex functions, called locally and globally discrete midpoint convex functions. These functions enjoy nice structural properties. They are stable under scaling and addition and satisfy a family of inequalities named parallelogram inequalities. Furthermore, they admit a proximity theorem with the same small proximity bound as that for L-(sic)-convex functions. These structural properties allow us to develop an algorithm for the minimization of locally and globally discrete midpoint convex functions based on the proximity-scaling approach and on a novel 2-neighborhood steepest descent algorithm.

关键词： midpoint convexity discrete convex function integral convexity L-(sic)-convexity proximity theorem scaling algorithm

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