Skip to main content
SpringerLink
Log in

Complexity and nonlinear semidefinite programming reformulation of \(\ell _1\)-constrained nonconvex quadratic optimization

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

Abstract

In this article, we show that the \(\ell _1\)-constrained nonconvex quadratic optimization problem (QPL1) is NP-hard, even when the off-diagonal elements are all nonnegative. Then we give an answer to Pinar and Teboulle’s open problem on the nonlinear semidefinite programming relaxation of (QPL1). The analytical approach is extended to \(\ell _p\)-constrained quadratic programs with \(1<p<2\).

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

Similar content being viewed by others

Notes

  1. The conjecture was raised in the personal communication with Professor Shu-Cherng Fang on December 2012.

References

  1. Bhaskara, A., Vijayaraghavan, A.: Approximating matrix \(p\)-norms. In: Proceedings of the 22 annual ACM-SIAM symposium on discrete algorithms, pp. 497–511, San Francisco (2011)

  2. Bomze, I.M., Frommlet, F., Rubey, F.: Improved SDP bounds for minimizing quadratic functions over the \(\ell ^1\)-ball. Optim. Lett. 1, 49–59 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bomze, I.M., Locatelli, M., Tardella, F.: New and old bounds for standard quadratic optimization: dominance, equivalence and incomparability. Math. Program. Ser. A 115, 31–64 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  4. Khot, S., Naor, A.: Grothendieck-type inequalities in combinatorial optimization. Commun. Pure Appl. Math. 65(7), 992–1035 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  5. Kindler, G., Naor, A., Schechtman, G.: The UGC hardness threshold of the Grothendieck problem. Math. Oper. Res. 35(2), 267–283 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  6. Lovasz, L., Schrijver, A.: Cones of matrices and set-functions and 0–1 optimization. SIAM. J. Optim. 1, 166–190 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  7. Luss, R., Teboulle, M.: Convex approximations to sparse PCA via Lagrangian duality. Oper. Res. Lett. 39(1), 57–61 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  8. Nesterov, Y.: Global quadratic optimization via conic relaxation. In: Wolkowicz, H., Saigal, R., Vandenberghe, L. (eds.) Handbook of Semidefinite Programming, pp. 363–384. Kluwer Academic Publishers, Boston (2000)

    Google Scholar 

  9. Pinar, M.Ç., Teboulle, M.: On semidefinite bounds for maximization of a non-convex quadratic objective over the \(\ell _1\) unit ball. RAIRO Oper. Res. 40(3), 253–265 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  10. Steinberg, D.: Computation of matrix norms with applications to robust optimization. Masters thesis, Technion–Israel Institute of Technology, Technion (2005)

  11. Sturm, J.F., Zhang, S.: On cones of nonnegative quadratic functions. Math. Oper. Res. 28, 246–267 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  12. Zhang, S.: Quadratic maximization and semidefinite relaxation. Math. Prog. 87, 453–465 (2000)

    Article  MATH  Google Scholar 

Download references

Acknowledgments

This research was supported by National Natural Science Foundation of China under Grants 11001006 and 91130019/A011702, and by the fund of State Key Laboratory of Software Development Environment under Grant SKLSDE-2013ZX-13. The author is grateful to the two anonymous referees for their valuable comments and suggestions that have greatly helped the author improve the paper.

Author information

Authors and Affiliations

  1. State Key Laboratory of Software Development Environment, LMIB of the Ministry of Education, School of Mathematics and System Sciences, Beihang University, Beijing, 100191, People’s Republic of China

    Yong Hsia

Authors
  1. Yong Hsia
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yong Hsia.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hsia, Y. Complexity and nonlinear semidefinite programming reformulation of \(\ell _1\)-constrained nonconvex quadratic optimization. Optim Lett 8, 1433–1442 (2014). https://doi.org/10.1007/s11590-013-0670-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11590-013-0670-1

Keywords

Mathematics Subject Classification (2010)

Search

Navigation