Journal of Global Optimization

, Volume 58, Issue 4, pp 631–652 | Cite as

Efficient computation of spectral bounds for Hessian matrices on hyperrectangles for global optimization

  • Moritz Schulze Darup
  • Martin Kastsian
  • Stefan Mross
  • Martin Mönnigmann


   We compare two established and a new method for the calculation of spectral bounds for Hessian matrices on hyperrectangles by applying them to a large collection of 1,522 objective and constraint functions extracted from benchmark global optimization problems. Both the tightness of the spectral bounds and the computational effort of the three methods, which apply to \(C^2\) functions \({\varphi }:\mathbb{R }^n\rightarrow \mathbb{R }\) that can be written as codelists, are assessed. Specifically, we compare eigenvalue bounds obtained with the interval variant of Gershgorin’s circle criterion (Adjiman et al. in Comput Chem Eng 22(9):1137–1158, 1998; Gershgorin in Izv. Akad. Nauk SSSR, Ser. fizmat. 6:749–754, 1931), Hertz (IEEE Trans Autom Control 37:532–535, 1992) and Rohn’s (SIAM J Matrix Anal Appl 15(1):175–184, 1994) method for tight bounds of interval matrices, and a recently proposed Hessian matrix eigenvalue arithmetic (Mönnigmann in SIAM J. Matrix Anal. Appl. 32(4): 1351–1366, 2011), which deliberately avoids the computation of interval Hessians. The eigenvalue arithmetic provides tighter, as tight, and less tight bounds than the interval variant of Gershgorin’s circle criterion in about 15, 61, and 24 % of the examples, respectively. Hertz and Rohn’s method results in bounds that are always as tight as or tighter than those from Gershgorin’s circle criterion, and as tight as or tighter than those from the eigenvalue arithmetic in 96 % of the cases. In 4 % of the examples, the eigenvalue arithmetic results in tighter bounds than Hertz and Rohn’s method. This result is surprising, since Hertz and Rohn’s method provides tight bounds for interval matrices. The eigenvalue arithmetic provides tighter bounds in these cases, since it is not based on interval matrices.


Eigenvalue bounds Spectral bounds Hessian Interval matrix  Global optimization 


  1. 1.
    Adjiman, C.S., Androulakis, I.P., Floudas, C.A.: A global optimization method, \(\alpha \)BB, for general twice-differentiabe constrained NLPs-II. Implementation and computational results. Comput. Chem. Eng. 22(9), 1159–1179 (1998)CrossRefGoogle Scholar
  2. 2.
    Adjiman, C.S., Dallwig, S., Floudas, C.A., Neumaier, A.: A global optimization method, \(\alpha \)BB, for general twice-differentiable constrained NLPs-I. Theoritical advances. Comput. Chem. Eng. 22(9), 1137–1158 (1998)CrossRefGoogle Scholar
  3. 3.
    Androulakis, I., Maranas, C., Floudas, C.A.: \(\alpha \)BB: a global optimization method for general constrained nonconvex problems. J. Glob. Opt. 7(4), 337–363 (1995)CrossRefGoogle Scholar
  4. 4.
    Boyd, S., Vandenberghe, L.: Convex Optimization. Cambrige University Press, Cambrige (2004)CrossRefGoogle Scholar
  5. 5.
    Corliss, G., Faure, C., Griewank, A., Hascoet, L., Naumann, U. (eds.): Automatic Differentiation of Algorithms: From Simulation to Optimization. Springer, Berlin (2002)Google Scholar
  6. 6.
    Du, K., Kearfott, R.: The cluster problem in multivariate global optimization. J. Glob. Opt. 5(3), 253–265 (1994)CrossRefGoogle Scholar
  7. 7.
    Fischer, H.: Automatisches Differenzieren. In: J. Herzberger (ed.) Wissenschaftliches Rechnen: eine Einführung in das Scientific Computing, pp. 53–103. Akademie, Berlin (1995)Google Scholar
  8. 8.
    Gershgorin, S.: Über die Abgrenzung der Eigenwerte einer Matrix. Izv. Akad. Nauk SSSR, Ser. fizmat. 6, 749–754 (1931)Google Scholar
  9. 9.
    Hertz, D.: The extreme eigenvalues and stability of real symmetric interval matrices. IEEE Trans. Autom. Cont. 37, 532–535 (1992)CrossRefGoogle Scholar
  10. 10.
    Hladík, M., Daney, D., Tsigaridas, E.: Bounds on real eigenvalues and singular values of interval matrices. SIAM J. Matrix Anal. Appl. 31(4), 2116–2129 (2010)CrossRefGoogle Scholar
  11. 11.
    Maranas, C., Floudas, C.A.: A deterministic global optimization approach for molecular-structure determination. J. Chem. Phys. 100(2), 1247–1261 (1994)CrossRefGoogle Scholar
  12. 12.
    Maranas, C., Floudas, C.A.: Global minimum potential energy conformations of small molecules. J. Glob. Opt. 4(2), 135–170 (1994)CrossRefGoogle Scholar
  13. 13.
    McCormick, G.: Computability of global solutions of factorable nonconvex programs-1 convex understimating problems. Math. Program. 10(2), 147–175 (1976)CrossRefGoogle Scholar
  14. 14.
    Mönnigmann, M.: Efficient calculation of bounds on spectra of Hessian matrices. SIAM J. Sci. Comput. 30, 2340–2357 (2008)CrossRefGoogle Scholar
  15. 15.
    Mönnigmann, M.: Positive invariance tests with efficient Hessian matrix eigenvalue bounds. In: Proceedings of 17th IFAC World Congress (2008)Google Scholar
  16. 16.
    Mönnigmann, M.: Fast calculation of spectral bounds for Hessian matrices on hyperrectangles. SIAM J. Matrix Anal. Appl. 32(4), 1351–1366 (2011)Google Scholar
  17. 17.
    Neumaier, A.: Interval Methods for Systems of Equations, Encyclopedia of Mathematics and its Applications, 1st edn. Cambrige University Press, Cambrige (2008)Google Scholar
  18. 18.
    Rall, L.B.: Automatic Differentiation: Techniques and Applications, Lecture Notes in Computer Science, vol. 120. Springer, Berlin (1981)CrossRefGoogle Scholar
  19. 19.
    Rockafellar, R.: Convex Analysis. Princeton University Press, Princeton (1997)Google Scholar
  20. 20.
    Rohn, J.: Positive definiteness and stability of interval matrices. SIAM J. Matrix Anal. Appl. 15(1), 175–184 (1994)CrossRefGoogle Scholar
  21. 21.
    Schichl, H., Markot, M.: Algorithmic differentiation techniques for global optimization in the coconut environment. J. Opt. Methods Softw. 27(2), 359–372 (2012)CrossRefGoogle Scholar
  22. 22.
    Shcherbina, O., Neumaier, A., Sam-Haroud, D., Vu, X.H., Nguyen, T.V.: Benchmarking global optimization and constraint satisfaction codes. In: Bliek, C., Jermann, C., Neumaier, A. (eds.) Global Optimization and Constraint Satisfaction, pp. 211–222. Springer, Berlin (2003)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Moritz Schulze Darup
    • 1
  • Martin Kastsian
    • 1
  • Stefan Mross
    • 1
  • Martin Mönnigmann
    • 1
  1. 1.Automatic Control and Systems TheoryRuhr-Universität BochumBochumGermany

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