Convexity Properties Associated with Nonconvex Quadratic Matrix Functions and Applications to Quadratic Programming



We establish several convexity results which are concerned with nonconvex quadratic matrix (QM) functions: strong duality of quadratic matrix programming problems, convexity of the image of mappings comprised of several QM functions and existence of a corresponding S-lemma. As a consequence of our results, we prove that a class of quadratic problems involving several functions with similar matrix terms has a zero duality gap. We present applications to robust optimization, to solution of linear systems immune to implementation errors and to the problem of computing the Chebyshev center of an intersection of balls.


Quadratic matrix functions Strong duality Extended S-lemma Semidefinite relaxation Convexity of quadratic maps 


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  1. 1.
    Beck, A.: Quadratic matrix programming. SIAM J. Optim. 17(4), 1224–1238 (2006) CrossRefMathSciNetGoogle Scholar
  2. 2.
    Jakubovič, V.A.: The S-procedure in nonlinear control theory. Vestn. Leningr. Univ. 1, 62–77 (1971) Google Scholar
  3. 3.
    Fradkov, A.L., Yakubovich, V.A.: The S-procedure and the duality relation in convex quadratic programming problems. Vestn. Leningr. Univ. 155(1), 81–87 (1973) Google Scholar
  4. 4.
    Polyak, B.T.: Convexity of quadratic transformations and its use in control and optimization. J. Optim. Theory Appl. 99(3), 553–583 (1998) MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Ye, Y., Zhang, S.: New results on quadratic minimization. SIAM J. Optim. 14, 245–267 (2003) MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Moré, J.J., Sorensen, D.C.: Computing a trust region step. SIAM J. Sci. Statist. Comput. 4(3), 553–572 (1983) MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Moré, J.J.: Generalization of the trust region problem. Optim. Methods Softw. 2, 189–209 (1993) CrossRefGoogle Scholar
  8. 8.
    Ben-Tal, A., Teboulle, M.: Hidden convexity in some nonconvex quadratically constrained quadratic programming. Math. Program. 72(1), 51–63 (1996) CrossRefMathSciNetGoogle Scholar
  9. 9.
    Fortin, C., Wolkowicz, H.: The trust region subproblem and semidefinite programming. Optim. Methods Softw. 19(1), 41–67 (2004) MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Stern, R.J., Wolkowicz, H.: Indefinite trust region subproblems and nonsymmetric eigenvalue perturbations. SIAM J. Optim. 5(2), 286–313 (1995) MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Beck, A., Eldar, Y.C.: Strong duality in nonconvex quadratic optimization with two quadratic constraints. SIAM J. Optim. 17(3), 844–860 (2006) MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Huang, Y., Zhang, S.: Complex matrix decomposition and quadratic programming. Technical Report (2005) Google Scholar
  13. 13.
    Pólik, I., Terlaky, T.: S-lemma: a survey. SIAM Rev. 49(3), 371–418 (2007) MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Ben-Tal, A., Nemirovski, A.: Lectures on Modern Convex Optimization. MPS-SIAM Series on Optimization. SIAM, Philadelphia (2001) MATHGoogle Scholar
  15. 15.
    Hiriart-Urruty, J.B., Torki, M.: Permanently going back and forth between the “quadratic world” and the “convexity world” in optimization. Appl. Math. Optim. 45(2), 169–184 (2002) MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Au-Yeung, Y.H., Poon, Y.T.: A remark on the convexity and positive definiteness concerning Hermitian matrices. Southeast Asian Bull. Math. 3(2), 85–92 (1979) MATHMathSciNetGoogle Scholar
  17. 17.
    Beck, A.: On the convexity of a class of quadratic mappings and its application to the problem of finding the smallest ball enclosing a given intersection of balls. J. Glob. Optim. 39(1), 113–126 (2007) MATHCrossRefGoogle Scholar
  18. 18.
    Pataki, G.: The geometry of semidefinite programming. In: Handbook of Semidefinite Programming. Internat. Ser. Oper. Res. Management Sci., vol. 27, pp. 29–65. Kluwer Academic, Dordrecht (2000) Google Scholar
  19. 19.
    Pataki, G.: On the rank of extreme matrices in semidefinite programs and the multiplicity of optimal eigenvalues. Math. Oper. Res. 23(2), 339–358 (1998) MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Barvinok, A.: A remark on the rank of positive semidefinite matrices subject to affine constraints. Discrete Comput. Geom. 25(1), 23–31 (2001) MATHMathSciNetGoogle Scholar
  21. 21.
    Van Huffel, S., Vandewalle, J.: The Total Least-Squares Problem: Computational Aspects and Analysis. Frontier in Applied Mathematics, vol. 9. SIAM, Philadelphia (1991) MATHGoogle Scholar
  22. 22.
    Guo, Y., Levy, B.C.: Worst-case MSE precoder design for imperfectly known MIMO communications channels. IEEE Trans. Signal Process. 53(8), 2918–2930 (2005) CrossRefMathSciNetGoogle Scholar
  23. 23.
    Ben-Tal, A., Nemirovski, A.: Robust convex optimization. Math. Oper. Res. 23(4), 769–805 (1998) MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Traub, J.F., Wasilkowski, G., Woźniakowski, H.: Information-Based Complexity. Computer Science and Scientific Computing. Academic Press, San Diego (1988). With contributions by A.G. Werschulz and T. Boult MATHGoogle Scholar
  25. 25.
    Xu, S., Freund, R.M., Sun, J.: Solution methodologies for the smallest enclosing circle problem. Comput. Optim. Appl. 25(1–3), 283–292 (2003). Atribute to Elijah (Lucien) Polak MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Brickman, L.: On the field of values of a matrix. Proc. Am. Math. Soc. 12, 61–66 (1961) MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Rockafellar, R.T.: Convex Analysis. Princeton Mathematical Series, vol. 28. Princeton University Press, Princeton (1970) MATHGoogle Scholar

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© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Faculty of Industrial Engineering and ManagementTechnion–Israel Institute of TechnologyHaifaIsrael

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