TY - JOUR

T1 - Discrete midpoint convexity

AU - Moriguchi, Satoko

AU - Murota, Kazuo

AU - Tamura, Akihisa

AU - Tardella, Fabio

N1 - Funding Information:
Funding: Financial support was received by the Japan Society for the Promotion of Science [JSPS KAKENHI JP16K00023, JSPS KAKENHI JP17K00037, JSPS KAKENHI JP24300003, JSPS KAKENHI JP26280004, JSPS KAKENHI JP26350430], the Mitsubishi Foundation, and Core Research for Evolutional Science and Technology, Japan Science and Technology Agency [JPMJCR14D2].
Publisher Copyright:
© 2019 INFORMS.

PY - 2020/2

Y1 - 2020/2

N2 - 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-convexity, and submodularity. By considering discrete midpoint convexity for all pairs at ℓ∞-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-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.

AB - 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-convexity, and submodularity. By considering discrete midpoint convexity for all pairs at ℓ∞-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-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.

KW - Discrete convex function

KW - Integral convexity

KW - L-convexity

KW - Midpoint convexity

KW - Proximity theorem

KW - Scaling algorithm

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U2 - 10.1287/MOOR.2018.0984

DO - 10.1287/MOOR.2018.0984

M3 - Article

AN - SCOPUS:85075196096

VL - 45

SP - 99

EP - 128

JO - Mathematics of Operations Research

JF - Mathematics of Operations Research

SN - 0364-765X

IS - 1

ER -