Skip to main content
SpringerLink
Log in

Efficient local search procedures for quadratic fractional programming problems

  • Published:
Computational Optimization and Applications Aims and scope Submit manuscript

Abstract

The problem of minimizing the sum of a convex quadratic function and the ratio of two quadratic functions can be reformulated as a Celis–Dennis–Tapia (CDT) problem and, thus, according to some recent results, can be polynomially solved. However, the degree of the known polynomial approaches for these problems is fairly large and that justifies the search for efficient local search procedures. In this paper the CDT reformulation of the problem is exploited to define a local search algorithm. On the theoretical side, its convergence to a stationary point is proved. On the practical side it is shown, through different numerical experiments, that the main cost of the algorithm is a single Schur decomposition to be performed during the initialization phase. The theoretical and practical results for this algorithm are further strengthened in a special case.

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
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

References

  1. Beck, A., Ben-Tal, A.: On the solution of the Tikhonov regularization of the total least squares problem. SIAM J. Optim. 17(1), 98–118 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  2. Beck, A., Ben-Tal, A., Teboulle, M.: 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 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  3. 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)

    Article  MathSciNet  MATH  Google Scholar 

  4. Beck, A., Teboulle, M.: On minimizing quadratically constrained ratio of two quadratic functions. J. Convex Anal. 17, 789–804 (2010)

    MathSciNet  MATH  Google Scholar 

  5. Benson, H.P.: Global optimization of nonlinear sums of ratios. J. Math. Anal. Appl. 263, 301–315 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  6. Benson, H.P.: Global optimization algorithm for the nonlinear sum of ratios problem. J. Optim. Theory Appl. 112, 1–29 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  7. Benson, H.P.: Using concave envelopes to globally solve the nonlinear sum of ratios problems. J. Glob. Optim. 22, 343–364 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  8. Benson, H.P.: Solving sum of ratios fractional programs via concave minimization. J. Optim. Theory Appl. 135, 1–17 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  9. Ben-Tal, A., den Hertog, D.: Hidden conic quadratic representation of some nonconvex quadratic optimization problems. Math. Program. 143, 1–29 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bienstock, D.: A note on polynomial solvability of the CDT problem. SIAM J. Optim. 26(1), 488–498 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  11. Celis, M.R., Dennis, J.E., Tapia, R.A.: A trust region strategy for nonlinear equality constrained optimization. In: Boggs, P.T., Byrd, R.H., Schnabel, R.B. (eds.) Numerical Optimization, pp. 71–82. SIAM, Philadelphia (1985)

    Google Scholar 

  12. Consolini, L., Locatelli, M.: On the complexity of quadratic programming with two quadratic constraints. Math. Program. 164(1–2), 91–128 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  13. Depetrini, D., Locatelli, M.: Approximation of linear fractional/multiplicative problems. Math. Program. 128, 437–443 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. Dinkelbach, W.: On nonlinear fractional programming. Manag. Sci. 13, 492–498 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  15. Fierro, R.D., Golub, G.H., Hansen, P.C., O’Leary, D.P.: Regularization by truncated total least squares. SIAM J. Sci. Comput. 18(4), 1223–1241 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  16. Freund, R.W., Jarre, F.: Solving the sum-of-ratios problem by an interior-point method. J. Glob. Optim. 19, 83–102 (2001)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  18. Hansen, P.C., O’Leary, D.P.: The use of the L-curve in the regularization of discrete ill-posed problems. SIAM J. Sci. Comput. 14(6), 1487–1503 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  19. Hansen, P.C.: Regularization tools: a Matlab package for analysis and solution of discrete ill-posed problems. Numer. Algorithms 6(1), 1–35 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  20. Hansen, P.C.: Deconvolution and regularization with Toeplitz matrices. Numer. Algorithms 29, 323–378 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  21. Konno, H., Fukaishi, K.: A branch and bound algorithm for solving low rank linear multiplicative and fractional programming problems. J. Glob. Optim. 18, 283–299 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kuno, T.: A branch-and-bound algorithm for maximizing the sum of several linear ratios. J. Glob. Optim. 22, 155–174 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  23. Lampe, J., Voss, H.: Large-scale Tikhonov regularization of total least squares. J. Comput. Appl. Math. 238, 95–108 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  24. Locatelli, M.: Alternative branching rules for some nonconvex problems. Optim. Methods Softw. 30(2), 365–378 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  25. Lu, S., Pereverzev, S.V., Tautenhahn, U.: Regularized total least squares: computational aspects and error bounds. SIAM J. Matrix Anal. Appl. 31(3), 918–941 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  26. Matsui, T.: NP-hardness of linear multiplicative programming and related problems. J. Glob. Optim. 9, 113–119 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  27. Mittal, S., Schulz, A.S.: An FPTAS for optimizing a class of low-rank functions over a polytope. Math. Program. 141, 103–120 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  28. Nguyen, V.-B., Sheu, R.-L., Xia, Y.: An SDP approach for quadratic fractional problems with a two-sided quadratic constraint. Optim. Methods Softw. 31(4), 701–719 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  29. Sakaue, S., Nakatsukasa, Y., Takeda, A., Iwata, S.: Solving generalized CDT problems via two-parameter eigenvalues. SIAM J. Optim. 26(3), 1669–1694 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  30. Schaible, S.: Fractional programming. In: Horst, R., Pardalos, P. (eds.) Handbook of Global Optimization, vol. 1. Kluwer Academic Publishers, Berlin (1995)

    Google Scholar 

  31. Schaible, S., Shi, J.: Fractional programming: the sum-of-ratios case. Optim. Methods Softw. 18, 219–229 (2003)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  33. Uhlig, F.: Definite and semidefinite matrices in a real symmetric matrix pencil. Pac. J. Math. 49, 561–568 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  34. Wang, Y.-J., Zhang, K.-C.: Global optimization of nonlinear sum of ratios problem. Appl. Math. Comput. 158(2), 319–330 (2004)

    MathSciNet  MATH  Google Scholar 

  35. Yang, M., Xia, Y., Wang, J., Peng, J.: Efficiently solving total least squares with Tikhonov identical regularization. Comput. Optim. Appl. 70, 571–592 (2018)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This research was supported by NSFC under Grants 11571029, 11471325 and 11771056.

Author information

Authors and Affiliations

  1. Dipartimento di Ingegneria e Architettura, Università di Parma, Parco Area delle Scienze, 181/A, 43124, Parma, Italy

    Luca Consolini & Marco Locatelli

  2. LMIB of the Ministry of Education, School of Mathematics and System Sciences, Beihang University, Beijing, 100191, People’s Republic of China

    Jiulin Wang & Yong Xia

Authors
  1. Luca Consolini
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Marco Locatelli
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jiulin Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Yong Xia
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Marco 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

Consolini, L., Locatelli, M., Wang, J. et al. Efficient local search procedures for quadratic fractional programming problems. Comput Optim Appl 76, 201–232 (2020). https://doi.org/10.1007/s10589-020-00175-1

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10589-020-00175-1

Keywords

Search

Navigation