Skip to main content
SpringerLink
Log in

A computational study on QP problems with general linear constraints

  • Original Paper
  • Published:
Optimization Letters Aims and scope Submit manuscript

Abstract

In this paper we consider Quadratic Programming (QP) problems with general linear constraints. We show, through a computational investigation, that a careful selection of a suitable reformulation of such problems, together with the related relaxation, and an intensive application of bound tightening are simple but very effective ingredients in order to make a standard branch and bound approach very competitive and in some cases able to outperform even well known commercial solvers.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

References

  1. Bonami, P., Günlük, O., Linderoth, J.: Globally solving nonconvex quadratic programming problems with box constraints via integer programming methods. Math. Program. Comput. 10, 333–382 (2018)

    Article  MathSciNet  Google Scholar 

  2. Burer, S., Vandenbussche, D.: Globally solving box-constrained non-convex quadratic programs with semidefinite-based finite branch-and-bound. Comput. Optim. Appl. 43(2), 181–195 (2009)

    Article  MathSciNet  Google Scholar 

  3. Caprara, A., Locatelli, M.: Global optimization problems and domain reduction strategies. Math. Program. 125(1), 123–137 (2010)

    Article  MathSciNet  Google Scholar 

  4. Caprara, A., Locatelli, M., Monaci, M.: Theoretical and computational results about optimality-based domain reductions. Comput. Optim. Appl. 64, 513–533 (2016)

    Article  MathSciNet  Google Scholar 

  5. Chen, J., Burer, S.: Globally solving nonconvex quadratic programming problems via completely positive programming. Math. Program. Comput. 4(1), 33–52 (2012)

    Article  MathSciNet  Google Scholar 

  6. Furini, F., Traversi, E., Belotti, P., Frangioni, A., Gleixner, A., Gould, N., Liberti, L., Lodi, A., Misener, R., Mittelmann, H., Sahinidis, N.V., Vigerske, S., Wiegele, A.: QPLIB: a library of quadratic programming instances. Math. Program. Comput. 11(2), 237–265 (2019)

    Article  MathSciNet  Google Scholar 

  7. Giannessi, F., Tomasin, E.: Nonconvex quadratic programs, linear complementarity problems, and integer linear programs, Lecture Notes in Computer Science, vol. 3, Springer, Berlin, pp. 437–449 (1973)

  8. Gleixner, A., Berthold, T., Müller, B., Weltge, S.: Three enhancements for optimization-based bound tightening. J. Global Optim. 67, 731–757 (2017)

    Article  MathSciNet  Google Scholar 

  9. Gondzio, J., Yildrim, E.A.: Global solutions of nonconvex Standard Quadratic Programs via Mixed Integer Linear Programming reformulations, available at https://arxiv.org/pdf/1810.02307.pdf

  10. Hansen, P., Jaumard, B., Ruiz, M., Xiong, J.: Global minimization of indefinite quadratic functions subject to box constraints. Nav. Res. Logist. 40(3), 373–392 (1993)

    Article  MathSciNet  Google Scholar 

  11. Liuzzi, G., Locatelli, M., Piccialli, V.: A new branch-and-bound algorithm for standard quadratic programming problems. Optim. Methods Softw. 34, 79–97 (2019)

    Article  MathSciNet  Google Scholar 

  12. Liuzzi, G., Locatelli, M., Piccialli, V., Rass, S.: Computing mixed strategies equilibria in presence of switching costs by the solution of nonconvex QP problems, Computational OPtimization and Applications, to appear, available at (2021) http://www.optimization-online.org/DB_FILE/2020/03/7656.pdf

  13. McCormick, G.P.: Computability of global solutions to factorable nonconvex programs: part I-convex underestimating problems. Math. Program. 10, 147–175 (1976)

    Article  Google Scholar 

  14. Motzkin, T.S., Straus, E.G.: Maxima for graphs and a new proof of a theorem of Turán. Canadian J. Math. 17(4), 533–540 (1965)

    Article  MathSciNet  Google Scholar 

  15. Nohra, C.J., Raghunathan, A.U., Sahinidis, N.V.: Spectral relaxations and branching strategies for global optimization of mixed-integer quadratic programs. SIAM J. Optim. 31, 142–171 (2021)

    Article  MathSciNet  Google Scholar 

  16. Pardalos, P.M., Vavasis, S.A.: Quadratic programming with one negative eigenvalue is NP-hard. J. Global Optim. 1(1), 15–22 (1991)

    Article  MathSciNet  Google Scholar 

  17. Sahinidis, N.V.: BARON: a general purpose global optimization software package. J. Global Optim. 8(2), 201–205 (1996)

    Article  MathSciNet  Google Scholar 

  18. Tawarmalani, M., Sahinidis, N.V.: Global optimization of mixed-integer nonlinear programs: a theoretical andcomputational study. Math. Program. 99(3), 563–591 (2004)

    Article  MathSciNet  Google Scholar 

  19. Xia, W., Vera, J.C., Zuluaga, L.F.: Globally solving nonconvex quadratic programs via linear integer programming techniques. INFORMS J. Comput. 32(1), 40–56 (2020)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. DIAG - “Sapienza” University of Rome/IASI, CNR, Rome, Italy

    G. Liuzzi

  2. DIA, Università degli Studi di Parma, Parma, Italy

    M. Locatelli

  3. DICII - University of Rome Tor Vergata, Rome, Italy

    V. Piccialli

Authors
  1. G. Liuzzi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. Locatelli
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. V. Piccialli
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. Locatelli.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liuzzi, G., Locatelli, M. & Piccialli, V. A computational study on QP problems with general linear constraints. Optim Lett 16, 1633–1647 (2022). https://doi.org/10.1007/s11590-021-01846-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11590-021-01846-6

Keywords

Search

Navigation