Mathematical Programming

, Volume 152, Issue 1–2, pp 435–466 | Cite as

A framework of discrete DC programming by discrete convex analysis

  • Takanori Maehara
  • Kazuo Murota
Full Length Paper Series A


A theoretical framework of difference of discrete convex functions (discrete DC functions) and optimization problems for discrete DC functions is established. Standard results in continuous DC theory are exported to the discrete DC theory with the use of discrete convex analysis. A discrete DC algorithm, which is a discrete analogue of the continuous DC algorithm (Concave–Convex procedure in machine learning) is proposed. The algorithm contains the submodular-supermodular procedure as a special case. Exploiting the polyhedral structure of discrete convex functions, the algorithms tailored to specific types of discrete DC functions are proposed.

Mathematics Subject Classification

90C27 90C46 



The authors thank Satoru Iwata and Akiyoshi Shioura for helpful discussions, Tom McCormick and Maurice Queyranne for communicating references [43]. This work is supported by KAKENHI (21360045, 26280004) and the Aihara Project, the FIRST program from JSPS.


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Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2014

Authors and Affiliations

  1. 1.JST, ERATO, Kawarabayashi ProjectNational Institute of InformaticsTokyoJapan
  2. 2.Department of Mathematical InformaticsUniversity of TokyoTokyoJapan

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