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Cone superadditivity of discrete convex functions

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Abstract

A function f is said to be cone superadditive if there exists a partition of R n into a family of polyhedral convex cones such that f(z + x) + f(z + y) ≤ f(z) + f(z + x + y) holds whenever x and y belong to the same cone in the family. This concept is useful in nonlinear integer programming in that, if the objective function is cone superadditive, the global minimality can be characterized by local optimality criterion involving Hilbert bases. This paper shows cone superadditivity of L-convex and M-convex functions with respect to conic partitions that are independent of particular functions. L-convex and M-convex functions in discrete variables (integer vectors) as well as in continuous variables (real vectors) are considered.

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Authors and Affiliations

  1. Department of Mathematical Informatics, Graduate School of Information Science and Technology, University of Tokyo, Tokyo, 113-8656, Japan

    Yusuke Kobayashi & Kazuo Murota

  2. Department of Mathematics, IFOR, ETH Zürich, Rämistrasse 101, 8092, Zürich, Switzerland

    Robert Weismantel

Authors
  1. Yusuke Kobayashi
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  2. Kazuo Murota
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  3. Robert Weismantel
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Correspondence to Yusuke Kobayashi.

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Kobayashi, Y., Murota, K. & Weismantel, R. Cone superadditivity of discrete convex functions. Math. Program. 135, 25–44 (2012). https://doi.org/10.1007/s10107-011-0447-1

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  • DOI: https://doi.org/10.1007/s10107-011-0447-1

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