Nothing Special   »   [go: up one dir, main page]
Skip to main content
Springer Nature Link
Log in

Solving the complex quadratic double-ratio minimax optimization under a quadratic constraint

  • Original Research
  • Published:
Journal of Applied Mathematics and Computing Aims and scope Submit manuscript

Abstract

Complex quadratic double-ratio minimax optimization (CQRMO) problem under a quadratic constraint has the potential to solve the total least squares problem. In order to solve it, a variant of S-Lemma is proposed and found to be interesting because it leads to a generalized linear conic fractional problem. Then, we achieve the global optimum of CQRMO problem with a quadratic constraint by using two algorithms for the generalized linear conic fractional problem. The efficiency of the proposed algorithms is evaluated by several numerical examples.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Ben-Tal, A., Nemirovski, A.: Lectures on modern convex optimization: analysis, algorithms, and engineering applications, SIAM, (2001)

  2. Blaimer, M., Heim, M., Neumann, D., Jakob, P.M., Kannengiesser, S., Breuer, F.: Comparison of phase constrained parallel MRI approaches: analogies and differences 75(3), 1086–1099 (2016)

  3. Chalise, B.K., Vandendrope, L.: Optimization of MIMO relays for multipoint-to-multipoint communications: Nonrobust and robust designs. IEEE Trans. Signal Process. 58(12), 6355–6368 (2010)

    Article  MATH  Google Scholar 

  4. Chen, J., Al-Homidan, S., Ansari, Q.H., Li, J., Lv, Y.: Robust necessary optimality conditions for nondifferentiable complex fractional programming with uncertain data. J. Optim. Theory Appl. 189(1), 221–243 (2021)

    Article  MATH  Google Scholar 

  5. Crouzeix, J.P., Ferland, J.A., Schaible, S.: An algorithm for generalized fractional programs. J. Optim. Theory Appl. 47(1), 35–49 (1985)

    Article  MATH  Google Scholar 

  6. Crouzeix, J.P., Ferland, J.A., Schaible, S.: A note on an algorithm for generalized fractional programs. J. Optim. Theory Appl. 50(1), 183–187 (1986)

    Article  MATH  Google Scholar 

  7. Datta, N., Bhatia, D.: Duality for a class of nondifferentiable mathematical programming problems in complex space. J. Math. Anal. Appl. 101(1), 1–11 (1984)

    Article  MATH  Google Scholar 

  8. K. Fan, Minimax theorems, Proceedings of the National Academy of Sciences of the United States of America, 39(1) (1953) 42-47

  9. Flachs, J.: Generalized Cheney-Loeb-Dinkelbach-type algorithms. Math. Oper. Res. 10(4), 674–687 (1985)

    Article  MATH  Google Scholar 

  10. Fradkov, A., Yakubovich, V.: The S-procedure and the duality relation in convex quadratic programming problems. Vestnik Leningrad. Univ 1(1), 81–87 (1973)

    MATH  Google Scholar 

  11. Grant, M., Boyd, S.: CVX: MATLAB Software for Disciplined Convex Programming, Version 2.1, http://cvxr.com/cvx (2015)

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

    Article  MATH  Google Scholar 

  13. Huang, T.Y.: Second-order parametric free dualities for complex minimax fractional programming. Mathematics 8(1), 67–79 (2020)

    Article  Google Scholar 

  14. Husain, Z., Ahmad, I., Sharma, S.: Second order duality for minmax fractional programming. Optim. Lett. 3(2), 277–286 (2009)

    Article  MATH  Google Scholar 

  15. Kramer, G., Gastpar, M., Gupta, P.: Cooperative strategies and capacity theorem for relay networks. IEEE Trans. Inf. Theory 51(9), 3037–3063 (2005)

    Article  MATH  Google Scholar 

  16. Lai, HCh., Huang, T.Y.: Nondifferentiable minimax fractional programming in complex spaces with parametric duality. J. Global Optim. 53(2), 243–254 (2012)

    Article  MATH  Google Scholar 

  17. Laneman, J. N., Wornell, G. W., Tse, D. N. C.: An efficient protocol for realizing cooperative diversity in wireless networks. In: Proceedings IEEE International Symposium on Information Theory, Washington, DC, (2001)

  18. Neumann, J.V.: Zur theorie der gesellschaftsspiele. Math. Ann. 100, 295–320 (1928)

    Article  MATH  Google Scholar 

  19. Rashid, U., Tuan, H.D., Kha, H.H., Nguyen, H.H.: Joint optimization of source precoding and relay beamforming in wireless MIMO relay networks. IEEE Trans. Commun. 62(2), 488–499 (2014)

    Article  Google Scholar 

  20. Sheu, R. L.: An SDP approach for some quadratic fractional problems (Nonlinear Analysis and Convex Analysis)

  21. Singh, V.P., Chaturvedi, A.K.: Max-Min fairness based linear Transceiver-Relay design for MIMO interference relay channel. IET Commun. 11(9), 1485–1496 (2017)

    Article  Google Scholar 

  22. Verma, R.U., Zalmaim, G.J.: Second-order parametric optimality conditions in discrete minmax fractional programming. Comm. Appl. Nonlinear Anal. 23(3), 1–32 (2016)

    Google Scholar 

  23. Yang, X.M., Hou, S.H.: On minimax fractional optimality and duality with generalized convexity. J. Global Optim. 31(2), 235–252 (2005)

    Article  MATH  Google Scholar 

  24. Yuan, D.H., Liu, X.L., Chinchuluun, A., Pardalos, P.M.: Nondifferentiable minimax fractional programming problems with (C, \(\alpha \), \(\rho \), d)-convexity. J. Optim. Theory Appl. 129(1), 185–199 (2006)

    Article  MATH  Google Scholar 

  25. Zare, A., Ashrafi, A., Xia, Y.: Quadratic Double-Ratio minimax optimization. Oper. Res. Lett. 49(4), 543–547 (2021)

    Article  MATH  Google Scholar 

  26. Zhang, G., Li, Q., Zhang, Q., Qin, J., Yang, L.: Signal-to-interference-plus-noise ratio-based multi-relay beamforming for multi-user multiple-input multiple-output cognitive relay networks with interference from primary network. IET Commun. 9(2), 227–238 (2015)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Mathematics, Statistics and Computer Science, Semnan University, Semnan, Iran

    Arezu Zare

Authors
  1. Arezu Zare
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Arezu Zare.

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

Zare, A. Solving the complex quadratic double-ratio minimax optimization under a quadratic constraint. J. Appl. Math. Comput. 69, 589–602 (2023). https://doi.org/10.1007/s12190-022-01762-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12190-022-01762-7

Keywords

Mathematics Subject Classification

Search

Navigation