Journal of Global Optimization

, Volume 73, Issue 2, pp 349–369 | Cite as

Geometric properties for level sets of quadratic functions

  • Huu-Quang Nguyen
  • Ruey-Lin SheuEmail author


In this paper, we study some fundamental geometrical properties related to the \({\mathcal {S}}\)-procedure. Given a pair of quadratic functions (gf), it asks when “\(g(x)=0 \Longrightarrow ~ f(x)\ge 0\)” can imply “(\(\exists \lambda \in {\mathbb {R}}\)) (\(\forall x\in {\mathbb {R}}^n\)\(f(x) + \lambda g(x)\ge 0.\)” Although the question has been answered by Xia et al. (Math Program 156:513–547, 2016), we propose a neat geometric proof for it (see Theorem 2): the \({\mathcal {S}}\)-procedure holds when, and only when, the level set \(\{g=0\}\) cannot separate the sublevel set \(\{f<0\}.\) With such a separation property, we proceed to prove that, for two polynomials (gf) both of degree 2, the image set of g over \(\{f<0\}, g(\{f<0\})\), is always connected (see Theorem 4). It implies that the \({\mathcal {S}}\)-procedure is a kind of the intermediate value theorem. As a consequence, we know not only the infimum of g over \(\{f\le 0\}\), but the extended results when g over \(\{f\le 0\}\) is unbounded from below or bounded but unattainable. The robustness and the sensitivity analysis of an optimization problem involving the pair (gf) automatically follows. All the results in this paper are novel and fundamental in control theory and optimization.


\({\mathcal {S}}\)-procedure Separation property S-lemma with equality Slater condition Intermediate value theorem Control theory 



Funding was provided by Ministry of Science and Technology, Taiwan (Grant No. MOST 105-2115-M-006-005-MY2).


  1. 1.
    Albert, A.: Conditions for positive and nonnegative definiteness in terms of pseudoinverses. SIAM J. Appl. Math. 17, 434–440 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Anstreicher, K.M., Wright, M.H.: A note on the augmented Hessian when the reduced Hessian is semidefinite. SIAM J. Optim. 11, 243–253 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Beck, A., Eldar, Y.C.: Strong duality in nonconvex quadratic optimization with two quadratic constraint. SIAM J. Optim. 17, 844–860 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Boyd, S.: Linear Matrix Inequalities in System and Control Theory, vol. 15. SIAM, Philadelphia (1994)CrossRefzbMATHGoogle Scholar
  5. 5.
    Derinkuyu, K., Pinar, M.C.: On the S-procedure and some variants. Math. Methods Oper. Res. 64, 55–77 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Dines, L.L.: On the mapping of quadratic forms. Bull. Am. Math. Soc. 47, 494–498 (1941)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Finsler, P.: Über das Vorkommen definiter und semidefiniter Formen und Scharen quadratischer Formen. Comment. Math. Helv. 9, 188–192 (1937)CrossRefzbMATHGoogle Scholar
  8. 8.
    Fradkov, A.L., Yakubovich, V.A.: The \({\cal{S}}\)-procedure and the duality relations in nonconvex problems of quadratic programmming. Vestnik Leningrad. Univ. Math. 6, 101–109 (1979)zbMATHGoogle Scholar
  9. 9.
    Hsia, Y., Lin, G.X., Sheu, R.L.: A revisit to quadratic programming with one inequality quadratic constraint via matrix pencil. Pac. J. Optim. 10, 461–481 (2014)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Luo, Z.Q., Sturm, J.F., Zhang, S.: Multivariate nonnegative quadratic mappings. SIAM J. Optim. 14, 1149–1162 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Moré, J.J.: Generalizations of the trust region problem. Optim. Methods Softw. 2, 189–209 (1993)CrossRefGoogle Scholar
  12. 12.
    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, 701–719 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Polik I, I., Terlaky, T.: A survey of the S-lemma. SIAM Rev. 49, 371–418 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Polyak, B.T.: Convexity of quadratic transformations and its use in control and optimization. J. Optim. Theory Appl. 99, 553–583 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Xia, Y., Wang, S., Sheu, R.L.: S-lemma with equality and its applications. Math. Program. 156, 513–547 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Tuy, H., Tuan, H.D.: Generalized S-lemma and strong duality in nonconvex quadratic programming. J. Glob. Optim. 56, 1045–1072 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Yakubovich, V.A.: The \({\cal{S}}\)-procedure in non-linear control theory. Vestnik Leningrad. Univ. Math., 4, pp. 73–93 (1977) (in Russian 1971) Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of Natural Science EducationVinh UniversityVinhVietnam
  2. 2.Department of MathematicsNational Cheng Kung UniversityTainanTaiwan

Personalised recommendations