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Computational Optimization and Applications

, Volume 63, Issue 3, pp 705–735 | Cite as

Approximated perspective relaxations: a project and lift approach

  • Antonio Frangioni
  • Fabio Furini
  • Claudio Gentile
Article

Abstract

The perspective reformulation (PR) of a Mixed-Integer NonLinear Program with semi-continuous variables is obtained by replacing each term in the (separable) objective function with its convex envelope. Solving the corresponding continuous relaxation requires appropriate techniques. Under some rather restrictive assumptions, the Projected PR (\(\mathrm{P}^2\mathrm{R}\)) can be defined where the integer variables are eliminated by projecting the solution set onto the space of the continuous variables only. This approach produces a simple piecewise-convex problem with the same structure as the original one; however, this prevents the use of general-purpose solvers, in that some variables are then only implicitly represented in the formulation. We show how to construct an Approximated Projected PR (\(\mathrm{AP}^2\mathrm{R}\)) whereby the projected formulation is “lifted” back to the original variable space, with each integer variable expressing one piece of the obtained piecewise-convex function. In some cases, this produces a reformulation of the original problem with exactly the same size and structure as the standard continuous relaxation, but providing substantially improved bounds. In the process we also substantially extend the approach beyond the original \(\mathrm{P}^2\mathrm{R}\) development by relaxing the requirement that the objective function be quadratic and the left endpoint of the domain of the variables be non-negative. While the \(\mathrm{AP}^2\mathrm{R}\) bound can be weaker than that of the PR, this approach can be applied in many more cases and allows direct use of off-the-shelf MINLP software; this is shown to be competitive with previously proposed approaches in some applications.

Keywords

Mixed-integer nonlinear problems Semi-continuous variables Perspective reformulation Projection 

Mathematics Subject Classification

90C06 90C25 

Notes

Acknowledgments

The first and third authors gratefully acknowledge the contribution of the Italian Ministry for University and Research under the PRIN 2012 Project 2012JXB3YF “Mixed-Integer Nonlinear Optimization: Approaches and Applications”, as well as of the European Union under the 7FP Marie Curie Initial Training Network n. 316647 “MINO: Mixed-Integer Nonlinear Optimization”.

References

  1. 1.
    Abhishek, K., Leyffer, S., Linderoth, J.: FilMINT: an outer-approximation-based solver for nonlinear mixed integer programs. INFORMS J. Comput. 22, 555–567 (2010)CrossRefMathSciNetMATHGoogle Scholar
  2. 2.
    Adams, W.P., Sherali, H.D.: A tight linearization and an algorithm for zero-one quadratic programming problems. Manag. Sci. 32(10), 1274–1290 (1986)CrossRefMathSciNetMATHGoogle Scholar
  3. 3.
    Adams, W.P., Sherali, H.D.: Linearization strategies for a class of zero-one mixed integer programming problems. Op. Res. 38(2), 217–226 (1990)CrossRefMathSciNetMATHGoogle Scholar
  4. 4.
    Adams, W.P., Sherali, H.D.: Mixed-integer bilinear programming problems. Math. Progr. 59(3), 279–306 (1993)CrossRefMathSciNetMATHGoogle Scholar
  5. 5.
    Agnetis, A., Grande, E., Pacifici, A.: Demand allocation with latency cost functions. Math. Progr. 132(1–2), 277–294 (2012)CrossRefMathSciNetMATHGoogle Scholar
  6. 6.
    Aktürk, S., Atamtürk, A., Gürel, S.: A strong conic quadratic reformulation for machine-job assignment with controllable processing times. Op. Res. Lett. 37(3), 187–191 (2009)CrossRefMATHGoogle Scholar
  7. 7.
    Bonami, P., Kilinç, M., Linderoth, J.: Algorithms and software for convex mixed integer nonlinear programs. In: S. Leyffer J. Lee, (ed.), Mixed Integer Nonlinear Programming, volume 154 of The IMA Volumes in Mathematics and its Applications, pages 61–89. (2012)Google Scholar
  8. 8.
    Ceria, S., Soares, J.: Convex programming for disjunctive convex optimization. Math. Progr. 86, 595–614 (1999)CrossRefMathSciNetMATHGoogle Scholar
  9. 9.
    Cui, X., Zheng, X., Zhu, S., Sun, X.: Convex relaxations and MIQCQP reformulations for a class of cardinality-constrained portfolio selection problems. J. Glob. Optim. 56(4), 1409–1423 (2012)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Frangioni, A., Galli, L., Scutellà, M.G.: Delay-constrained shortest paths: approximation algorithms and second-order cone models. J. Optim. Theory Appl. 164(3), 1051–1077 (2015)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Frangioni, A., Gentile, C.: Perspective cuts for 0–1 mixed integer programs. Math. Progr. 106(2), 225–236 (2006)CrossRefMathSciNetMATHGoogle Scholar
  12. 12.
    Frangioni, A., Gentile, C.: SDP diagonalizations and perspective cuts for a class of nonseparable MIQP. Op. Res. Lett. 35(2), 181–185 (2007)CrossRefMathSciNetMATHGoogle Scholar
  13. 13.
    Frangioni, A., Gentile, C.: A computational comparison of reformulations of the perspective relaxation: SOCP vs cutting planes. Op. Res. Lett. 37(3), 206–210 (2009)CrossRefMathSciNetMATHGoogle Scholar
  14. 14.
    Frangioni, A., Gentile, C., Grande, E., Pacifici, A.: Projected perspective reformulations with applications in design problems. Op. Res. 59(5), 1225–1232 (2011)CrossRefMathSciNetMATHGoogle Scholar
  15. 15.
    Frangioni, A., Gentile, C., Lacalandra, F.: Solving unit commitment problems with general ramp contraints. Int. J. Electr. Power Energy Syst. 30, 316–326 (2008)CrossRefGoogle Scholar
  16. 16.
    Frangioni, A., Gentile, C., Lacalandra, F.: Tighter approximated MILP formulations for unit commitment problems. IEEE Trans. Power Syst. 24(1), 105–113 (2009)CrossRefGoogle Scholar
  17. 17.
    Frangioni, A., Gentile, C., Lacalandra, F.: Sequential Lagrangian-MILP approaches for unit commitment problems. Int. J. Electr. Power Energy Syst. 33, 585–593 (2011)CrossRefGoogle Scholar
  18. 18.
    Frangioni, A., Gorgone, E.: A library for continuous convex separable quadratic knapsack problems. Eur. J. Oper. Res. 229(1), 37–40 (2013)CrossRefMathSciNetGoogle Scholar
  19. 19.
    Gounaris, C.E., Floudas, C.A.: Tight convex underestimators for \(C^2\)-continuous problems: I. univariate functions. J. Glob. Optim. 42, 51–67 (2008)CrossRefMathSciNetMATHGoogle Scholar
  20. 20.
    Günlük, O., Lee, J., Weismantel, R.: MINLP Strengthening for Separable Convex Quadratic Transportation-Cost UFL. IBM Research Report RC24213, IBM Research Division, (2007)Google Scholar
  21. 21.
    Günlük, O., Linderoth, J.: Perspective relaxation of MINLPs with indicator variables. In: A. Lodi, A. Panconesi, and G. Rinaldi. (eds.), Proceedings \(13th\) IPCO, volume 5035 of Lecture Notes in Computer Science, pp. 1–16, (2008)Google Scholar
  22. 22.
    Günlük, O., Linderoth, J.: Perspective reformulation and applications. In: S. Leyffer J. Lee. (ed.), Mixed Integer Nonlinear Programming, volume 154 of The IMA Volumes in Mathematics and its Applications, pp. 61–89. (2012)Google Scholar
  23. 23.
    Hijazi, H., Bonami, P., Cornuejols, G., Ouorou, A.: Mixed integer nonlinear programs featuring “on/off” constraints: convex analysis and applications. Electron. Notes Discret. Math. 36(1), 1153–1160 (2010)CrossRefGoogle Scholar
  24. 24.
    Khajavirad, A., Sahinidis, N.V.: Convex envelopes generated from finitely many compact convex sets. Math. Progr. 137(1–2), 371–408 (2013)CrossRefMathSciNetMATHGoogle Scholar
  25. 25.
    Lemaréchal, C., Ouorou, A., Petrou, G.: A bundle-type algorithm for routing in telecommunication data networks. Comput. Optim. Appl. 44(3), 385–409 (2009)CrossRefMathSciNetMATHGoogle Scholar
  26. 26.
    Leyffer, S.: Experiments with MINLP branching techniques. In: European Workshop on Mixed Integer Nonlinear Programming, (2010)Google Scholar
  27. 27.
    Stubbs, R.A., Mehrotra, S.: A branch-and-cut method for 0–1 mixed convex programming. Math. Progr. 86, 515–532 (1999)CrossRefMathSciNetMATHGoogle Scholar
  28. 28.
    Tahanan, M., van Ackooij, W., Frangioni, A., Lacalandra, F.: Large-scale Unit Commitment under uncertainty. 4OR 13(2), 115–171 (2015)CrossRefMathSciNetGoogle Scholar
  29. 29.
    Tawarmalani, M., Richard, J.-P.P., Xiong, C.: Explicit convex and concave envelopes through polyhedral subdivisions. Math. Progr. 138, 531–577 (2013)CrossRefMathSciNetMATHGoogle Scholar
  30. 30.
    Tawarmalani, M., Sahinidis, N.V.: A polyhedral branch-and-cut approach to global optimization. Math. Progr. 103, 225–249 (2005)CrossRefMathSciNetMATHGoogle Scholar
  31. 31.
    Tawarmalani, M., Sahinidis, N.V.: Semidefinite relaxations of fractional programs via novel convexification techniques. J. Glob. Optim. 20, 133–154 (2001)CrossRefMathSciNetGoogle Scholar
  32. 32.
    Tawarmalani, M., Sahinidis, N.V.: Convex extensions and envelopes of lower semi-continuous functions. Math. Progr. 93, 515–532 (2002)CrossRefMathSciNetGoogle Scholar
  33. 33.
    Zamora, J.M., Grossmann, I.E.: A global mINLP optimization algorithm for the synthesis of heat exchanger networks with no stream splits. Comput. Chem. Eng. 22, 367–384 (1998)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Antonio Frangioni
    • 1
  • Fabio Furini
    • 2
  • Claudio Gentile
    • 3
  1. 1.Dipartimento di InformaticaUniversità di PisaPisaItaly
  2. 2.LAMSADEUniversité Paris-DauphineParisFrance
  3. 3.Istituto di Analisi dei Sistemi ed Informatica “A. Ruberti” (IASI-CNR), Consiglio Nazionale delle RicercheRomeItaly

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