TY - JOUR

T1 - Scaling, proximity, and optimization of integrally convex functions

AU - Moriguchi, Satoko

AU - Murota, Kazuo

AU - Tamura, Akihisa

AU - Tardella, Fabio

N1 - Funding Information:
Acknowledgements The authors thank Yoshio Okamoto for communicating a relevant reference and two anonymous referees for detailed comments. This research was initiated at the Trimester Program “Combinatorial Optimization” at Hausdorff Institute of Mathematics, 2015. This work was supported by The Mitsubishi Foundation, CREST, JST, Grant Number JPMJCR14D2, Japan, and JSPS KAKENHI Grant Numbers JP26350430, JP26280004, JP16K00023, JP17K00037.
Publisher Copyright:
© 2018, Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society.

PY - 2019/5/1

Y1 - 2019/5/1

N2 - In discrete convex analysis, the scaling and proximity properties for the class of L ♮ -convex functions were established more than a decade ago and have been used to design efficient minimization algorithms. For the larger class of integrally convex functions of n variables, we show here that the scaling property only holds when n≤ 2 , while a proximity theorem can be established for any n, but only with a superexponential bound. This is, however, sufficient to extend the classical logarithmic complexity result for minimizing a discrete convex function of one variable to the case of integrally convex functions of any fixed number of variables.

AB - In discrete convex analysis, the scaling and proximity properties for the class of L ♮ -convex functions were established more than a decade ago and have been used to design efficient minimization algorithms. For the larger class of integrally convex functions of n variables, we show here that the scaling property only holds when n≤ 2 , while a proximity theorem can be established for any n, but only with a superexponential bound. This is, however, sufficient to extend the classical logarithmic complexity result for minimizing a discrete convex function of one variable to the case of integrally convex functions of any fixed number of variables.

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U2 - 10.1007/s10107-018-1234-z

DO - 10.1007/s10107-018-1234-z

M3 - Article

AN - SCOPUS:85040931141

SN - 0025-5610

VL - 175

SP - 119

EP - 154

JO - Mathematical Programming

JF - Mathematical Programming

IS - 1-2

ER -