Skip to main content
SpringerLink
Log in

A convex optimization approach for minimizing the ratio of indefinite quadratic functions over an ellipsoid

  • FULL LENGTH PAPER
  • Published:
Mathematical Programming Submit manuscript

Abstract

We consider the nonconvex problem (RQ) of minimizing the ratio of two nonconvex quadratic functions over a possibly degenerate ellipsoid. This formulation is motivated by the so-called regularized total least squares problem (RTLS), which is a special case of the problem’s class we study. We prove that under a certain mild assumption on the problem’s data, problem (RQ) admits an exact semidefinite programming relaxation. We then study a simple iterative procedure which is proven to converge superlinearly to a global solution of (RQ) and show that the dependency of the number of iterations on the optimality tolerance \(\varepsilon\) grows as \(O(\sqrt{\ln \varepsilon^{-1}})\).

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

References

  1. Anstreicher K., Chen X., Wolkowicz H. and Yuan Y. (1999). Strong duality for a trust-region type relaxation of the quadratic assignment problem. Linear Algebra Appl. 301(1–3): 121–136

    Article  MATH  MathSciNet  Google Scholar 

  2. Anstreicher K. and Wolkowicz H. (2000). On Lagrangian relaxation of quadratic matrix constraints. SIAM J. Matrix Anal. Appl. 22(1): 41–55

    Article  MATH  MathSciNet  Google Scholar 

  3. Auslender A. and Teboulle M. (2003). Asymptotic Cones and Functions in Optimization and Variational Inequalities. Springer, Heidelberg

    MATH  Google Scholar 

  4. Beck A. (2006). Quadratic matrix programming. SIAM J. Optim. 17(4): 1224–1238

    Article  MathSciNet  Google Scholar 

  5. Beck A., Ben-Tal A. and Teboulle M. (2006). Finding a global optimal solution for a quadratically constrained fractional quadratic problem with applications to the regularized total least squares. SIAM J. Matrix Anal. Appl. 28(2): 425–445

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  7. Ben-Tal, A., Nemirovski, A.: Lectures on Modern Convex Optimization. MPS-SIAM Series on Optimization (2001)

  8. Ben-Tal A. and Teboulle M. (1996). Hidden convexity in some nonconvex quadratically constrained quadratic programming. Math. Program. 72(1): 51–63

    Article  MathSciNet  Google Scholar 

  9. Dinkelbach W. (1996). On nonlinear fractional programming. Math. Program. 13: 492–498

    MathSciNet  Google Scholar 

  10. Fortin C. and Wolkowicz H. (2004). The trust region subproblem and semidefinite programming. Optim. Methods Softw. 19(1): 41–67

    Article  MATH  MathSciNet  Google Scholar 

  11. Golub G.H., Hansen P.C. and O’Leary D.P. (1999). Tikhonov regularization and total least squares. SIAM J. Matrix Anal. Appl. 21(2): 185–194

    Article  MATH  MathSciNet  Google Scholar 

  12. Golub G.H. and Van Loan C.F. (1980). An analysis of the total least-squares problem. SIAM J. Numer. Anal. 17(6): 883–893

    Article  MATH  MathSciNet  Google Scholar 

  13. Guo, H., Renaut, R.: A regularized total least squares algorithm. In:Total Least Squares and Errors-in-Variables Modeling, pp. 57–66. Kluwer, Dordrecht (2002)

  14. Van Huffel, S., Vandewalle, J.: The total least-squares problem: computational aspects and analysis, vol. 9 of Frontier in applied mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1991)

  15. Moré J.J. (1993). Generalization of the trust region problem. Optim. Methods Softw. 2: 189–209

    Article  Google Scholar 

  16. Nesterov, Y., Nemirovskii, A.: Interior-point polynomial algorithms in convex programming, vol. 13 of SIAM Studies in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1994)

  17. Polyak B.T. (1998). Convexity of quadratic transformations and its use in control and optimization. J. Optim. Theory Appl. 99(3): 553–583

    Article  MATH  MathSciNet  Google Scholar 

  18. Renaut R.A. and Guo H. (2005). Efficient algorithms for solution of regularized total least squares. SIAM J. Matrix Anal. Appl. 26(2): 457–476

    Article  MATH  MathSciNet  Google Scholar 

  19. Rendl F. and Wolkowicz H. (1997). A semidefinite framework for trust region subproblems with applications to large scale minimization. Math. Program. 77(2, Ser.B): 273–299

    MathSciNet  Google Scholar 

  20. Sima D., Van Huffel S. and Golub G.H. (2004). Regularized total least squares based on quadratic eigenvalue problem solvers. BIT Numer. Math. 44(4): 793–812

    Article  MATH  MathSciNet  Google Scholar 

  21. Sorensen D.C. (1997). Minimization of a large-scale quadratic function subject to a spherical constraint. SIAM J. Optim. 7(1): 141–161

    Article  MATH  MathSciNet  Google Scholar 

  22. Stern R.J. and Wolkowicz H. (1995). Indefinite trust region subproblems and nonsymmetric eigenvalue perturbations. SIAM J. Optim. 5(2): 286–313

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Industrial Engineering, Technion, Israel Institute of Technology, Haifa, 32000, Israel

    Amir Beck

  2. School of Mathematical Sciences, Tel-Aviv University, Ramat-Aviv, Tel-Aviv, 69978, Israel

    Marc Teboulle

Authors
  1. Amir Beck
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Marc Teboulle
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Amir Beck.

Additional information

This research is partially supported by the Israel Science Foundation, ISF grant #489-06.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Beck, A., Teboulle, M. A convex optimization approach for minimizing the ratio of indefinite quadratic functions over an ellipsoid. Math. Program. 118, 13–35 (2009). https://doi.org/10.1007/s10107-007-0181-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10107-007-0181-x

Keywords

Mathematics Subject Classification (2000)

Search

Navigation