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Gaining or Losing Perspective

  • Jon LeeEmail author
  • Daphne Skipper
  • Emily Speakman
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 991)

Abstract

We study MINLO (mixed-integer nonlinear optimization) formulations of the disjunction \(x\in \{0\}\cup [l,u]\), where z is a binary indicator of \(x\in [l,u]\), and y “captures” \(x^p\), for \(p>1\). This model is useful when activities have operating ranges, we pay a fixed cost for carrying out each activity, and costs on the levels of activities are strictly convex. One well-known concrete application (with \(p=2\)) is mean-variance optimization (in the style of Markowitz).

Using volume as a measure to compare convex bodies, we investigate a family of relaxations for this model, employing the inequality \(yz^q \ge x^p\), parameterized by the “lifting exponent” \(q\in [0,p-1]\). These models are higher-dimensional-power-cone representable, and hence tractable in theory. We analytically determine the behavior of these relaxations as functions of lup and q. We validate our results computationally, for the case of \(p=2\). Furthermore, for \(p=2\), we obtain results on asymptotic behavior and on optimal branching-point selection.

Keywords

Mixed-integer nonlinear optimization Volume Integer Relaxation Polytope Perspective Higher-dimensional power cone 

References

  1. 1.
    Aktürk, M.S., Atamtürk, A., Gürel, S.: A strong conic quadratic reformulation for machine-job assignment with controllable processing times. Oper. Res. Lett. 37(3), 187–191 (2009)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Basu, A., Conforti, M., Di Summa, M., Zambelli, G.: Optimal cutting planes from the group relaxations. arXiv:abs/1710.07672 (2018)
  3. 3.
    Dey, S., Molinaro, M.: Theoretical challenges towards cutting-plane selection. arXiv:abs/1805.02782 (2018)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Frangioni, A., Gentile, C.: Perspective cuts for a class of convex 0–1 mixed integer programs. Math. Program. 106(2), 225–236 (2006)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Grant, M., Boyd, S.: CVX: Matlab software for disciplined convex programming, version 2.1, build 1123. http://cvxr.com/cvx (2017)
  6. 6.
    Günlük, O., Linderoth, J.: Perspective reformulations of mixed integer nonlinear programs with indicator variables. Math. Program. Ser. B 124, 183–205 (2010)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Ko, C.W., Lee, J., Steingrímsson, E.: The volume of relaxed Boolean-quadric and cut polytopes. Discret. Math. 163(1–3), 293–298 (1997)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Lee, J., Morris Jr., W.D.: Geometric comparison of combinatorial polytopes. Discret. Appl. Math. 55(2), 163–182 (1994)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Lee, J., Skipper, D.: Volume computation for sparse boolean quadric relaxations. Discret. Appl. Math. (2017).  https://doi.org/10.1016/j.dam.2018.10.038.
  10. 10.
    Speakman, E., Lee, J.: Quantifying double McCormick. Math. Oper. Res. 42(4), 1230–1253 (2017)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Speakman, E., Lee, J.: On branching-point selection for trilinear monomials in spatial branch-and-bound: the hull relaxation. J. Glob. Optim. (2018).  https://doi.org/10.1016/j.dam.2018.10.038.
  12. 12.
    Speakman, E., Yu, H., Lee, J.: Experimental validation of volume-based comparison for double-McCormick relaxations. In: Salvagnin, D., Lombardi, M. (eds.) CPAIOR 2017, pp. 229–243. Springer (2017)Google Scholar
  13. 13.
    Speakman, E.E.: Volumetric guidance for handling triple products in spatial branch-and-bound. Ph.D., University of Michigan (2017)Google Scholar
  14. 14.
    Steingrímsson, E.: A decomposition of \(2\)-weak vertex-packing polytopes. Discret. Comput. Geom. 12(4), 465–479 (1994)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Toh, K.C., Todd, M.J., Tütüncü, R.H.: SDPT3-a MATLAB software package for semidefinite programming. Optim. Methods Softw. 11, 545–581 (1998)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.University of MichiganAnn ArborUSA
  2. 2.U.S. Naval AcademyAnnapolisUSA
  3. 3.Otto-von-Guericke-UniversitätMagdeburgGermany

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