Skip to main content
SpringerLink
Log in

A branch and bound algorithm for nonconvex quadratic optimization with ball and linear constraints

  • Published:
Journal of Global Optimization Aims and scope Submit manuscript

Abstract

We suggest a branch and bound algorithm for solving continuous optimization problems where a (generally nonconvex) objective function is to be minimized under nonconvex inequality constraints which satisfy some specific solvability assumptions. The assumptions hold for some special cases of nonconvex quadratic optimization problems. We show how the algorithm can be applied to the problem of minimizing a nonconvex quadratic function under ball, out-of-ball and linear constraints. The main tool we utilize is the ability to solve in polynomial computation time the minimization of a general quadratic under one Euclidean sphere constraint, namely the so-called trust region subproblem, including the computation of all local minimizers of that problem. Application of the algorithm on sparse source localization problems is presented.

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
Fig. 2

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Notes

  1. That is, vectors which minimize the objective over a small neighborhood of the feasible domain, but are not global minimizers of (4.1) or (4.2).

References

  1. Achterberg, T., Berthold, T., Koch, T., Wolter, K.: Constraint integer programming: a new approach to integrate cp and mip. ZIB-Report 08-01 (2008)

  2. Beck, A., Eldar, Y.C.: Strong duality in nonconvex quadratic optimization with two quadratic constraints. SIAM J. Optim. 17(3), 844–860 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  3. Beck, A., Stoica, P., Li, J.: Exact and approximate solutions of source localiztaion problems. IEEE Trans. Signal Process. 56(5), 1770–1778 (2008)

    Article  MathSciNet  Google Scholar 

  4. Beck, A., Teboulle, M.: A fast iterative shrinkage-thresholding algorithm for linear invers problems. SIAM J. Imaging Sci. 2(1), 183–202 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Beck, A., Teboulle, M., Chikishev, Z.: Exact and approximate solutions of source localiztaion problems. IEEE Trans. Signal Process. 56(5), 1770–1778 (2008)

    Article  MathSciNet  Google Scholar 

  6. Bertsekas, D.P.: Nonlinear Programming, 2nd edn. Athena Scientific, Belmont (1999)

    MATH  Google Scholar 

  7. Bienstock, D., Michalka, A.: Polynomial solvability of variants of the trust-region subproblem. In: SODA, pp. 380–390 (2014)

  8. Bruckstein, A.M., Donoho, D.L., Elad, M.: From sparse solutions of systems of equations to sparse modeling of signals and images. SIAM Rev. 51(1), 34–81 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. Burer, S., Anstreicher, K.: Second-order cone constraints for extended trust-region subproblems. Manuscript, Department of Management Sciences, University of Iowa, Iowa City, IA 52242, (2011)

  10. Burer, S., Yang, B.: The Trust Region Subproblem with Non-intersecting Linear Constraints, Manuscript. University of Iowa, Iowa City (2013)

    MATH  Google Scholar 

  11. Chen, J., Burer, S.: Globally solving nonconvex quadratic programming problems via completely positive programming. Math. Program. Comput. 4, 33–52 (2012). doi:10.1007/s12532-011-0033-9

    Article  MathSciNet  MATH  Google Scholar 

  12. Cheung, K.W., Ma, W.K., So, H.C.: Accurate approximation algorithm for toa-based maximum likelihood mobile location using semidefinite programming. In: Proceedings of ICASSP, vol. 2, pp. 145–148 (2004)

  13. Gay, D.M.: Computing optimal locally constrained steps. SIAM J. Sci. Stat. Comput. 2(2), 186–197 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  14. Grant, M., Boyd, S.: Graph implementations for nonsmooth convex programs. In: Blondel, V., Boyd, S., Kimura, H. (eds.) Recent Advances in Learning and Control. Lecture Notes in Control and Information Sciences, pp. 95–110. Springer, Berlin (2008). http://stanford.edu/~boyd/graph_dcp.html

  15. Grant, M., Boyd, S.: CVX: Matlab software for disciplined convex programming, version 2.1, (March 2014) http://cvxr.com/cvx

  16. Hsia, Y., Sheu, R.L.: Trust region subproblem with a fixed number of additional linear inequality constraints has polynomial complexity (2013). arXiv:1312.1398

  17. Jeyakumar, V., Li, G.Y.: Trust-region problems with linear inequality constraints: exact sdp relaxation, global optimality and robust optimization. Math. Program. Ser. A 147, 171–206 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  18. Martínez, J.M.: Local minimizers of quadratic functions on euclidean balls and spheres. SIAM J. Optim. 4(1), 159–176 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  19. Moré, J.J.: Generalizations of the trust region subproblem. Optim. Methods Softw. 2, 189–209 (1993)

    Article  Google Scholar 

  20. Moré, J.J., Sorensen, D.C.: Computing a trust region step. SIAM J. Sci. Stat. Comput. 4(3), 553–572 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  21. Nocedal, J., Wright, S.J.: Numerical Optimization, 2nd edn. Springer, Berlin (2000)

    MATH  Google Scholar 

  22. Sorensen, D.C.: Newton’s method with a model trust region modification. SIAM J. Numer. Anal. 19(2), 409–426 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  23. Sturm, J., Zhang, S.: On cones of nonnegative quadratic functions. Math. Oper. Res. 288, 246–267 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  24. Ye, Y., Zhang, S.: New results on quadratic minimization. SIAM J. Optim. 14(1), 245–267 (2003)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The research of Amir Beck is partially supported by the Israel Science Foundation (ISF) Grant No. 1821/16.

Author information

Authors and Affiliations

  1. Faculty of Industrial Engineering and Management, Technion - Israel Institute of Technology, Haifa, Israel

    Amir Beck & Dror Pan

Authors
  1. Amir Beck
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dror Pan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Amir Beck.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Beck, A., Pan, D. A branch and bound algorithm for nonconvex quadratic optimization with ball and linear constraints. J Glob Optim 69, 309–342 (2017). https://doi.org/10.1007/s10898-017-0521-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-017-0521-1

Keywords

Search

Navigation