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Constructing lattice-free gradient polyhedra in dimension two

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Abstract

Lattice-free gradient polyhedra can be used to certify optimality for mixed integer convex minimization models. We consider how to construct these polyhedra for unconstrained models with two integer variables under the assumption that all level sets are bounded. In this setting, a classic result of Bell, Doignon, and Scarf states the existence of a lattice-free gradient polyhedron with at most four facets. We present an algorithm for creating a sequence of gradient polyhedra, each of which has at most four facets, that finitely converges to a lattice-free gradient polyhedron. Each update requires constantly many gradient evaluations. Our updates imitate the gradient descent algorithm, and consequently, it yields a gradient descent type of algorithm for problems with two integer variables.

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Notes

  1. We use ‘gradient evaluation’ to refer a single inner product evaluation using gradients, and ‘constantly many’ can be chosen to be 20 as counted in Table 1.

  2. Case 2 requires 6 oracle calls while the others require only 4. This is also somewhat of an overestimate as one gradient may be reused over the course of multiple iterations.

References

  1. Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)

    Book  Google Scholar 

  2. Bell, D.: A theorem concerning the integer lattice. Stud. Appl. Math. 56, 187–188 (1977)

    Article  MathSciNet  Google Scholar 

  3. Doignon, J.: Convexity in cristallographical lattices. J. Geom. 3, 71–85 (1973)

    Article  MathSciNet  Google Scholar 

  4. Scarf, H.: An observation on the structure of production sets with indivisibilities. Proc. Natl. Acad. Sci. U.S.A. 74, 3637–3641 (1977)

    Article  MathSciNet  Google Scholar 

  5. Baes, M., Oertel, T., Weismantel, R.: Duality for mixed-integer convex minimization. Math. Program. Ser. A 158, 547–564 (2016)

    Article  MathSciNet  Google Scholar 

  6. Basu, A., Conforti, M., Cornuéjols, G., Weismantel, R., Weltge, S.: Optimality certificates for convex minimization and Helly numbers. Op. Res. Let. 45, 671–674 (2017)

    Article  MathSciNet  Google Scholar 

  7. Andersen, K., Louveaux, Q., Weismantel, R., Wolsey, L.A.: Inequalities from two rows of a simplex tableau. In Fischetti M., Williamson D.P., eds.: Integer Programming and Combinatorial Optimization (IPCO). Volume 4513 of Lecture Notes in Computer Science., Springer, Berlin, Heidelberg (2007)

  8. Balas, E.: Intersection cuts - a new type of cutting planes for integer programming. Op. Res. 19(1), 19–39 (1971)

    Article  MathSciNet  Google Scholar 

  9. Basu, A., Conforti, M., Di Summa, M.: A geometric approach to cut-generating functions. Math. Program. 151(1), 153–189 (2015)

    Article  MathSciNet  Google Scholar 

  10. Basu, A., Cornuéjols, G., Köppe, M.: Unique minimal liftings for simplicial polytopes. Math. Op. Res. 37(2), 346–355 (2012)

    Article  MathSciNet  Google Scholar 

  11. Del Pia, A., Weismantel, R.: Relaxations of mixed integer sets from lattice-free polyhedra. 4OR 10(3), 221–244 (2012)

    Article  MathSciNet  Google Scholar 

  12. Averkov, G., Krümpelmann, J., Weltge, S.: Notions of maximality for integral lattice-free polyhedra: the case of dimension three. Mathematics of Operations Research 1035–1062 (2017)

  13. Averkov, G., Wagner, C., Weismantel, R.: Maximal lattice-free polyhedra: Finiteness and an explicit description in dimension three. Math. Oper. Res. 721–742 (2011)

  14. Basu, A., Conforti, M., Cornuéjols, G., Zambelli, G.: Maximal lattice-free convex sets in linear subspaces. Math. Op. Res. 35(3), 704–720 (2010)

    Article  MathSciNet  Google Scholar 

  15. Dey, S.S., Wolsey, L.A.: Lifting integer variables in minimal inequalities corresponding to lattice-free triangles. In Lodi A., Panconesi A., Rinaldi G., eds.: International Conference on Integer Programming and Combinatorial Optimization (IPCO). Volume 5035 of Lecture Notes in Computer Science., Springer, Berlin, Heidelberg (2008) 463 – 475

  16. Lovász, L.: Geometry of numbers and integer programming. In M.Iri, Tanabe, K., eds.: Mathematical Programming: Recent Developments and Applications, Kluwer Academic Publishers (1989) 177 – 201

  17. Gupta, O., Ravindran, V.: Branch and bound experiments in convex nonlinear integer programming. Manage. Sci. 31, 1533–1546 (1985)

    Article  MathSciNet  Google Scholar 

  18. Leyffer, S.: Integrating SQP and branch-and-bound for mixed integer nonlinear programming. Comput. Optim. Appl. 18, 295–309 (2001)

    Article  MathSciNet  Google Scholar 

  19. Bonami, P., Biegler, L.T., Conn, A.R., Cornuéjols, G., Grossman, I.E., Laird, C.D., Lee, J., Lodi, A., Margot, F., Sawaya, N., Wächter, A.: An algorithmic framework for convex mixed integer nonlinear programs. Discret. Opt. 5(2), 186–204 (2008)

    Article  MathSciNet  Google Scholar 

  20. Duran, M.A., Grossman, I.E.: An outer-approximation algorithm for a class of mixed-integer nonlinear programs. Math. Program. 36(3), 307–339 (1986)

    Article  MathSciNet  Google Scholar 

  21. Dadush, D., Peikert, C., Vempala, S.: Enumerative lattice algorithms in any norm via M-ellipsoid coverings. In: 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science. (2011) 580–589

  22. Grötschel, M., Lovász, L., Schrijver, A.: Geometric Algorithms and Combinatorial Optimization. 1 edn. Volume 2 of Algorithms and Combinatorics. Springer-Verlag Berlin Heidelberg (1988)

  23. Baes, M., Oertel, T., Wagner, C., Weismantel, R.: Mirror-descent methods in mixed-integer convex optimization. In: Jünger, M., Reinelt, G. (eds.) Facets Comb. Op., pp. 101–131. Springer, Berlin, Heidelberg (2013)

    Google Scholar 

  24. Barvinok, A.: A Course in Convexity. Volume 54. Graduate Studies in Mathematics, American Mathematical Society, Providence, Rhode Island (2002)

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Authors and Affiliations

  1. Sauder School of Business, University of British Columbia, Vancouver, BC, Canada

    Joseph Paat

  2. Department of Mathematics, ETH Zürich, Zürich, Switzerland

    Miriam Schlöter

  3. Department of Mathematical and Statistical Sciences, University of Colorado Denver, Denver, USA

    Emily Speakman

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  1. Joseph Paat
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  2. Miriam Schlöter
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  3. Emily Speakman
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Correspondence to Joseph Paat.

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This paper extends work from an IPCO paper of the same title. This version includes complete proofs as well as new results on convergence for functions that are Lipschitz continuous and strongly convex.

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Paat, J., Schlöter, M. & Speakman, E. Constructing lattice-free gradient polyhedra in dimension two. Math. Program. 192, 293–317 (2022). https://doi.org/10.1007/s10107-021-01658-7

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