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A framework of discrete DC programming by discrete convex analysis

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Abstract

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.

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Notes

  1. M\(^\natural \)-convex and L\(^\natural \)-convex functions are introduced by Murota and Shioura [38], and Fujishige and Murota [12], respectively, as variants of M-convex and L-convex functions; see [36]. M\(^\natural \)-concave functions on \(\{0,1\}^n\) are essentially valuated matroids of Dress and Wenzel [4]. We note that “M” stands for matroid, and “L” stands for lattice, and the symbol \(\natural \) is to read “natural.”

  2. The original definition [33] of matroid valuation on independent sets is a set function \(\zeta : 2^V \rightarrow \mathbb {Z} \cup \{-\infty \}\) that satisfies (i) \(\zeta (X)\) is not identically \(-\infty \), (ii) If \(X \subseteq Y\) then \(\zeta (X) \ge \zeta (Y)\), (iii) If \(X \subseteq Y\), \(|X| < |Y|\), and \(\zeta (Y) \ne -\infty \), there exists \(v \in V {\setminus } X\) such that \(\zeta (Y) = \zeta (Y + v)\), (iv) For \(X, Y \subseteq V\) and \(u \in X {\setminus } Y\), there exists \(v \in Y {\setminus } X\) such that \(\zeta (X) + \zeta (Y) \le \zeta (X - u + v) + \zeta (Y + u - v)\).

  3. Usually this theorem is used in combination of Hartman’s theorem [13] to prove the statement that “every \(C^2\) function is a continuous DC function.” See Hartman [13], Hiriart-Urruty [15] or Tuy [51].

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Acknowledgments

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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  1. JST, ERATO, Kawarabayashi Project, National Institute of Informatics, Tokyo, 101-8430, Japan

    Takanori Maehara

  2. Department of Mathematical Informatics, University of Tokyo, Tokyo, 113-8656, Japan

    Kazuo Murota

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  1. Takanori Maehara
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Correspondence to Takanori Maehara.

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Maehara, T., Murota, K. A framework of discrete DC programming by discrete convex analysis. Math. Program. 152, 435–466 (2015). https://doi.org/10.1007/s10107-014-0792-y

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