Solving a Type of the Tikhonov Regularization of the Total Least Squares by a New S-Lemma

  • Huu-Quang Nguyen
  • Ruey-Lin SheuEmail author
  • Yong Xia
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 991)


We present a new S-lemma with two quadratic equalities and use it to minimize a special type of polynomials of degree 4. As a result, by the Dinkelbach approach with 2 SDP’s (semidefinite programming), the minimum value and the minimum solution to the Tikhonov regularization of the total least squares problem with \(L=I\) can be nicely obtained.


S-lemma with equality Tikhonov regularization Total least squares Dinkelbach method 


  1. 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)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Beck, A., Eldar, Y.C.: Strong duality in nonconvex quadratic optimization with two quadratic constraint. SIAM J. Optim. 17(3), 844–860 (2006)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Derinkuyu, K., Pınar, M.Ç.: On the S-procedure and some variants. Math. Methods Oper. Res. 64(1), 55–77 (2006)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Dinkelbach, W.: On nonlinear fractional programming. Manag. Sci. 13, 492–498 (1967)MathSciNetCrossRefGoogle Scholar
  5. 5.
    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)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Polik, I., Terlaky, T.: A survey of the S-lemma. SIAM Rev. 49(3), 371–418 (2007)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Pong, T.K., Wolkowicz, H.: The generalized trust region subprobelm. Comput. Optim. Appl. 58, 273–322 (2014)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Polyak, B.T.: Convexity of quadratic transformations and its use in control and optimization. J. Optim. Theory Appl. 99(3), 553–583 (1998)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Rockefellar, R.T.: Convex Analysis. Princeton University Press (1970)Google Scholar
  10. 10.
    Stoer, J., Witzgall, C.: Convexity and Optimization in Finite Dimensions, vol. I. Springer-Verlag, Heidelberg (1970)CrossRefGoogle Scholar
  11. 11.
    Stern, R., Wolkowicz, H.: Indefinite trust region subproblems and nonsymmetric eigenvalue perturbations. SIAM J. Optim. 5(2), 286–313 (1995)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Wang, S., Xia, Y.: Strong duality for generalized trust region subproblem: S-lemma with interval bounds. Optim. Lett. 9(6), 1063–1073 (2015)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Xia, Y., Wang, S., Sheu, R.L.: S-lemma with equality and its applications. Math. Program. Ser. A. 156(1), 513–547 (2016)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Tuy, H., Tuan, H.D.: Generalized S-lemma and strong duality in nonconvex quadratic programming. J. Global Optim. 56, 1045–1072 (2013)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Yakubovich, V.A.: S-procedure in nonlinear control theory. Vestn. Leningr. Univ. 1, 62–77 (1971). (in Russian)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Yang, M., Yong, X., Wang, J., Peng, J.: Efficiently solving total least squares with Tikhonov identical regularization. Comput. Optim. Appl. 70(2), 571–592 (2018)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Department of MathematicsVinh UniversityVinhVietnam
  2. 2.Department of MathematicsNational Cheng Kung UniversityTainanTaiwan
  3. 3.State Key Laboratory of Software Development Environment School of Mathematics and System SciencesBeihang UniversityBeijingChina

Personalised recommendations