Journal of Global Optimization

, Volume 38, Issue 3, pp 479–501 | Cite as

A new class of test functions for global optimization

Original Paper


In this paper we propose a new class of test functions for unconstrained global optimization problems. The class depends on some parameters through which the difficulty of the test problems can be controlled. As a basis for future comparison, we propose a selected set of these functions, with increasing difficulty, and some computational experiments with two simple global optimization algorithms.


Global optimization Test problems Multilevel structure Molecular conformation problems 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Ackley D.H. (1987) A Connectionist Machine for Genetic Hillclimbing. Kluwer Academic Publishers, BostonGoogle Scholar
  2. Addis, B., Leyffer, S.: A trust-region algorithm for global optimization. Comput. Optim. Appl. (2004), in press.Google Scholar
  3. Addis, B., Locatelli, M., Schoen, F.: Local optima smoothing for global optimization. Optim. Method Softw. (2005), in press.Google Scholar
  4. Dixon, L.C.W., Szegö, G.P.: Towards Global Optimization 2. North-Holland, Amsterdam, The Netherlands (1978)Google Scholar
  5. Doye J.P.K. (2006) Physical perspectives on the global optimization of atomic clusters, to appear. In: Pintér J.D.(ed) Global Optimization: Scientific and Engineering Case Studies, Nonconvex Optimization and Its Applications Series: Vol. 85. Springer, BerlinGoogle Scholar
  6. Doye J.P.K., Leary R.H., Locatelli M., Schoen F. (2004) The global optimization of Morse clusters by potential transformations. INFORMS J. Comput. 16, 371–379CrossRefGoogle Scholar
  7. Floudas C.A., Pardalos P.M., Adjiman C.J., Esposito W.R., Gümüs Z.H., Harding S.T., Klepeis J.L., Meyer C.A., Schweiger C.A. (1999) Handbook of Test Problems in Local and Global Optimization, volume 33 of Nonconvex Optimization and its Applications. Kluwer Academic Publishers, DordrechtGoogle Scholar
  8. Floudas C.A., Jongen H.Th. (2005) Global optimization: local minima and transition points. J. Glob. Optim. 32, 409–415CrossRefGoogle Scholar
  9. Gaviano M., Kvasov D.E., Lera D., Sergeyev Y.D. (2003) Algorithm 829: software for generation of classes of test functions with known local and global minima for global optimization. ACM Trans. Math. Softw. 29, 469–480CrossRefGoogle Scholar
  10. Global Optimization web site by A. Neumaier, glopt/test.html (2005)Google Scholar
  11. Global Optimization Laboratory web site, TestFunctions1.0/index.html (2006)Google Scholar
  12. Hock W., Schittkowski K. (1981) Test examples for nonlinear programming codes. Lecture Notes in Economics and Mathematical Systems, Vol. 187. Springer, BerlinGoogle Scholar
  13. Kalantari B., Rosen J.B. (1986) Construction of large-scale global minimum concave quadratic test problems. J.Optim.Theory Appl. 48(2):303–313CrossRefGoogle Scholar
  14. Lavor C., Maculan N. (2004) A function to test methods applied to global minimization of potential energy of molecules. Num. Algorithms 35, 287–300CrossRefGoogle Scholar
  15. Leary R.H. (2000) Global optimization on funneling landscapes. J. Glob. Optim. 18, 367–383CrossRefGoogle Scholar
  16. Levy A., Montalvo A. (1985) The tunneling method for global optimization. SIAM J. Sci. Stat. Comp. 1, 15–29CrossRefGoogle Scholar
  17. Locatelli M. (2005) On the multilevel structure of global optimization problems. Comput. Optim. Appl. 30, 5–22CrossRefGoogle Scholar
  18. Mathar R., Žilinskas A. (1994) A class of test functions for global optimization. J. Glob. Optim. 5, 195–199CrossRefGoogle Scholar
  19. Moré J.J., Garbow B.S., Hillstrom K.E. (1981) Testing unconstrained optimization software. ACM Trans. Math. Softw. 7, 17–41CrossRefGoogle Scholar
  20. Neumaier A., Shcherbina O., Huyer W., Vinkó T. (2005) A comparison of complete global optimization solvers. Math. Program. 103(2):335–356CrossRefGoogle Scholar
  21. Pardalos P.M. (1987) Generation of large-scale quadratic programs for use as global optimization test problems. ACM Trans. Math. Soft. 13(2):133–137CrossRefGoogle Scholar
  22. Pintér J.D. (2002) Global optimization: software, test problems and applications. In: Pardalos P.M., Romeijn H.E. (eds) Handbook of Global Optimization, Volume 2. Kluwer Academic Publishers, Dordrecht, Boston, LondonGoogle Scholar
  23. RngStream class by Pierre L’Ecuyer, University of Montreal, (2004)Google Scholar
  24. Schittkowski K. (1987) More test examples for nonlinear programming. Lecture Notes in Economics and Mathematical Systems, vol. 282. Springer, BerlinGoogle Scholar
  25. Schoen F. (1993) A wide class of test functions for global optimization. J. Glob. Optim. 3, 133–137CrossRefGoogle Scholar
  26. Schwefel H.P. (1981) Numerical Optimization of Computer Models. WIley, ChicesterGoogle Scholar
  27. Törn A., Žilinskas A. (1989) Global Optimization. Lecture Notes in Computer Sciences. Springer-Verlag, BerlinGoogle Scholar
  28. Wales D.J., Doye J.P.K. (1997) Global optimization by basin-hopping and the lowest energy structures of Lennard-Jones clusters containing up to 110 atoms. J. Phys. Chem. A 101:5111–5116CrossRefGoogle Scholar
  29. Whitley D., Mathias K., Rana S., Dzubera J. (1995) Building better test functions. In: Eshelman (ed) Proceedings of the International Conference on Genetic Algorithms. Los Altos, CA, Morgan KaufmannGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  1. 1.Dip. Ingegneria dell’InformazioneUniversità di SienaSienaItaly
  2. 2.Dip. InformaticaUniversità di TorinoTorinoItaly

Personalised recommendations