Abstract
The infinite relaxations in integer programming were introduced by Gomory and Johnson to provide a general framework for the theory of cutting planes: the so-called valid functions, and in particular the minimal and extreme functions, can be seen as automatic rules for the generation of cuts. However, while many extreme functions are piecewise linear and therefore easy to describe, the set of extreme functions turns out to have a very complicated mathematical structure, as several extreme functions are known that exhibit a somewhat pathological behavior. In this paper we show that if some smoothness assumption is imposed on an extreme function \(\pi \), then \(\pi \) is necessarily piecewise linear. More precisely, we show that if a continuous extreme function for the Gomory–Johnson one-dimensional infinite group relaxation is a piecewise \({\mathcal {C}}^2\) function, then it is a piecewise linear function.
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Notes
Actually, the case \(\varDelta _\gamma (x,y)=0\) for every \(x,y\in [0,1]\) is not possible, as one can prove that in this situation \(\gamma (x)=0\) for every \(x\in [0,1]\), a contradiction to Lemma 18.
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The author is grateful to two anonymous referees and an associate editor, whose detailed comments were useful to improve the presentation of this paper.
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This work was supported by the Grant “SID 2016” of the University of Padova.
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Di Summa, M. Piecewise smooth extreme functions are piecewise linear. Math. Program. 179, 265–293 (2020). https://doi.org/10.1007/s10107-018-1330-0
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DOI: https://doi.org/10.1007/s10107-018-1330-0