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Light on the infinite group relaxation II: sufficient conditions for extremality, sequences, and algorithms

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Abstract

This is the second part of a survey on the infinite group problem, an infinite-dimensional relaxation of integer linear optimization problems introduced by Ralph Gomory and Ellis Johnson in their groundbreaking papers titled Some continuous functions related to corner polyhedra I, II (Math Program 3:23–85, 1972a; Math Program 3:359–389, 1972b). The survey presents the infinite group problem in the modern context of cut generating functions. It focuses on the recent developments, such as algorithms for testing extremality and breakthroughs for the k-row problem for general \(k\ge 1\) that extend previous work on the single-row and two-row problems. The survey also includes some previously unpublished results; among other things, it unveils piecewise linear extreme functions with more than four different slopes. An interactive companion program, implemented in the open-source computer algebra package Sage, provides an updated compendium of known extreme functions.

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Notes

  1. The function is available in the Electronic Compendium (Zhou 2014) as gmic.

  2. See Definition 3.7 (Basu et al. 2015, Part I).

  3. See the definition in (3.1) (Basu et al. 2015, Part I).

  4. This is a superadditive pseudo-periodic function in the terminology of Richard et al. (2009).

  5. See Sect. 3.6 (Basu et al. 2015, Part I).

  6. The statement of Proposition 6.1 remains true for generalizations of sequential limits; for example, we may consider the convergence of nets of minimal functions.

  7. The sequence and its limit can be constructed using drlm_gj_2_slope_extreme_limit_to_ nonextreme.

  8. See, for example, Hunter and Nachtergaele (2001) for an introduction to Sobolev spaces.

  9. These parameters are collected in the list delta, which is an argument to the function bhk_irrational. The parameters are \(\mathbb Q\)-linearly independent for example when one parameter is rational, e.g., 1/200 , the other irrational, e.g., sqrt(2)/200 . When the irrational number is algebraic (for example, when it is constructed using square roots), the code will construct an appropriate real number field that is a field extension of the rationals. In this field, the computations are done in exact arithmetic.

  10. Such a sequence and the limit can be constructed using bhk_irrational_extreme_limit_to_ rational_nonextreme.

  11. This can also be done with a finite left derivative. Note that not all extreme functions have a finite left or right derivative at the origin. That is, there exist extreme functions that are discontinuous on both sides of the origin. See Table 4 for examples.

  12. The first n terms of such a sequence of \(\epsilon _i\) are generated by e = generate_example_e_ for_psi_n(n= n).

  13. The construction of \(\psi _n\) is furnished by h = psi_n_in_bccz_counterexample_ construction(e=e), where e is the list [ \(\epsilon _1, \dots , \epsilon _n\) ].

  14. The function can be created by h = bccz_counterexample(); however, h(x) can be exactly evaluated only on the set \(\bigcup _{i=0}^\infty \hbox {cl}X_i^-\) defined below; for other values, the function will return an approximation.

  15. In fact, if \(\mu ^- < 1\), then \(\psi \) is actually Lipschitz continuous and thus absolutely continuous and hence almost everywhere differentiable. The convergence then holds even in the sense of the space \(W^{1,1}_{\mathrm {loc}}(\mathbb R)\).

  16. If h is the function \(\pi \), e.g., after typing h = dg_2_step_mir(), then the algorithm is invoked by typing extremality_test(h, show_plots=True). In the irrational case no proof of finite convergence of the procedure is known.

  17. Under these hypotheses, \(\pi \) is the continuous interpolation of \(\pi |_{\frac{1}{q}\mathbb Z}\).

  18. The function is available in the electronic compendium (Zhou 2014) as kzh_2q_example_1.

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Acknowledgments

Thanks go to Yuan Zhou for compiling the electronic compendium of extreme functions in Zhou (2014), and Chun Yu Hong and Yuan Zhou for their work on the software (Hong et al. 2014).

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Authors and Affiliations

  1. Department of Applied Mathematics and Statistics, The Johns Hopkins University, Baltimore, MD, 21218, USA

    Amitabh Basu

  2. Department of Mathematics, Institute for Operations Research, ETH Zürich, Zürich, Switzerland

    Robert Hildebrand

  3. Department of Mathematics, University of California, Davis, Davis, CA, 95616, USA

    Matthias Köppe

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  1. Amitabh Basu
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  2. Robert Hildebrand
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Correspondence to Matthias Köppe.

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The authors gratefully acknowledge partial support from the National Science Foundation through grants DMS-0914873 (R. Hildebrand, M. Köppe) and DMS-1320051 (M. Köppe).

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Basu, A., Hildebrand, R. & Köppe, M. Light on the infinite group relaxation II: sufficient conditions for extremality, sequences, and algorithms. 4OR-Q J Oper Res 14, 107–131 (2016). https://doi.org/10.1007/s10288-015-0293-8

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