Skip to main content
Log in

Bound constrained interval global optimization in the COCONUT Environment

  • Published:
Journal of Global Optimization Aims and scope Submit manuscript

Abstract

We introduce a new interval global optimization method for solving bound constrained problems. The method originates from a small standalone software and is implemented in the COCONUT Environment, a framework designed for the development of complex algorithms, containing numerous state-of-the-art methods in a common software platform. The original algorithm is enhanced by various new methods implemented in COCONUT, regarding both interval function evaluations (such as first and second order derivatives with backward automatic differentiation, slopes, slopes of derivatives, bicentered forms, evaluations on the Karush–John conditions, etc.) and algorithmic elements (inclusion/exclusion boxes, local search, constraint propagation). This resulted in a substantial performance increase as compared to the original code. During the selection of the best combination of options, we performed comparison tests that gave empirical answers to long-lasting algorithmic questions (such as whether to use interval gradients or use slopes instead), that have never been studied numerically in such detail before. The new algorithm, called coco_gop_ex, was tested against the prestigious BARON software on an extensive set of bound constrained problems. We found that in addition to accepting a wider class of bound constrained problems and providing more output information (by locating all global minimizers), coco_gop_ex is competitive with BARON in terms of the solution success rates (with the exception of a set of nonlinear least squares problems), and it often outperforms BARON in running time. In particular, coco_gop_ex was around 21 % faster on average over the set of problems solved by both software systems.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Baumann, E.: Optimal centered forms. BIT Numer. Math. 28, 80–87 (1988)

    Article  Google Scholar 

  2. Bliek, C.: Computer methods for design automation. PhD thesis, Department of Ocean Engineering, Massachusetts Institute of Technology (1992)

  3. Byrd, R.H., Lu, P., Nocedal, J., Zhu, C.: A limited memory algorithm for bound constrained optimization. SIAM J. Sci. Comput. 16, 1190–1208 (1995)

    Article  Google Scholar 

  4. COCONUT Environment. Software, University of Vienna. http://www.mat.univie.ac.at/coconut-environment

  5. Csendes, T., Ratz, D.: Subdivision direction selection in interval methods for global optimization. SIAM J. Numer. Anal. 34, 922–938 (1997)

    Article  Google Scholar 

  6. Gebremedhin, A.H., Manne, F., Pothen, A.: What color is your Jacobian? Graph coloring for computing derivatives. SIAM Rev. 47, 629–705 (2005)

    Article  Google Scholar 

  7. Griewank, A., Corliss, G.F.: Automatic Differentiation of Algorithms. SIAM, Philadelphia (1991)

    Google Scholar 

  8. Hammer, R., Hocks, M., Kulisch, U., Ratz, D.: C++ Toolbox for Verified Computing. Springer, Heidelberg (1995)

    Google Scholar 

  9. Kearfott, R.B., Du, K.: The cluster problem in multivariate global optimization. J. Glob. Optim. 5, 253–265 (1994)

    Article  Google Scholar 

  10. Kieffer, M., Markót, M.C., Schichl, H., Walter, E.: Verified global optimization for estimating the parameters of nonlinear models. In: Rauh, A., Auer, E. (eds.) Modeling, Design, and Simulation of Systems with Uncertainties, Chapter 7, Mathematical Engineering. Springer, New York (2011)

  11. Kolev, L.V.: Use of interval slopes for the irrational part of factorable functions. Reliab. Comput. 3, 83–93 (1997)

    Article  Google Scholar 

  12. Markót, M.C., Csendes, T.: A new verified optimization technique for the “packing circles in a unit square” problems. SIAM J. Optim. 16, 193–219 (2005)

    Article  Google Scholar 

  13. Markót, M.C., Csendes, T., Csallner, A.E.: Multisection in interval branch-and-bound methods for global optimization II. Numerical tests. J. Glob. Optim. 16, 219–228 (2000)

    Article  Google Scholar 

  14. Markót, M.C., Fernández, J., Casado, L.G., Csendes, T.: New interval methods for constrained global optimization. Math. Program. 106, 287–318 (2006)

    Article  Google Scholar 

  15. Markót, M.C., Schichl, H.: Comparison and automated selection of local optimization solvers for interval global optimization methods. SIAM J. Optim. 21, 1371–1391 (2011)

    Article  Google Scholar 

  16. Moore, R.E., Kearfott, R.B., Cloud, M.J.: Introduction to Interval Analysis. SIAM, Philadelphia (2009)

    Book  Google Scholar 

  17. Neumaier, A.: The enclosure of solutions of parameter-dependent systems of equations. In: Moore, R.E. (ed.) Reliability in Computing, pp. 269–286. Academic Press, San Diego (1988)

    Chapter  Google Scholar 

  18. Neumaier, A.: Interval Methods for Systems of Equations. Vol. 37 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (1990)

    Google Scholar 

  19. Ratschek, H.: Centered forms. SIAM J. Numer. Anal. 17, 656–662 (1980)

    Article  Google Scholar 

  20. Sahinidis, N.V., Tawarmalani, M.: BARON 9.0.4: global optimization of mixed-integer nonlinear programs, user’s manual, 2010. http://www.gams.com/dd/docs/solvers/baron.pdf

  21. Schichl, H., Neumaier, A.: Interval analysis on directed acyclic graphs for global optimization. J. Glob. Optim. 33, 541–562 (2005)

    Article  Google Scholar 

  22. Schichl, H.: Mathematical Modeling and Global Optimization. Habilitation Thesis, draft of a book, Cambridge University Press (to appear)

  23. Schichl, H., Neumaier, A.: Exclusion regions for systems of equations. SIAM J. Numer. Anal. 42, 383–408 (2004)

    Article  Google Scholar 

  24. Schichl, H., Markót, M.C.: Algorithmic differentiation techniques for global optimization in the COCONUT Environment. Optim. Methods Softw. 27, 359–372 (2012)

    Article  Google Scholar 

  25. Schichl, H., Markót, M.C., Neumaier, A.: Exclusion regions for optimization problems (in preparation)

  26. Tawarmalani, M., Sahinidis, N.V.: A polyhedral branch-and-cut approach to global optimization. Math. Program. 103, 225–249 (2005)

    Article  Google Scholar 

  27. Van Hentenryck, P., Numerica: a modeling language for global optimization. In: Proceedings of the 15th international joint conference on artificial intelligence (IJCAI-97) (1997)

  28. Vu, X.-H., Schichl, H., Sam-Haroud, D.: Interval propagation and search on directed acyclic graphs for numerical constraint solving. J. Global Optim. 45, 499–531 (2009)

    Article  Google Scholar 

Download references

Acknowledgments

We are grateful to Nick Sahinidis (Carnegie Mellon University) and to Michael R. Bussieck (GAMS) for providing us licenses to BARON and GAMS to carry out our numerical tests. We also thank Arnold Neumaier (University of Vienna) for his numerous suggestions during the development of our software.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Mihály Csaba Markót.

Additional information

This research was supported by the Austrian Science Found (FWF) Grants Nr. P18704-N13 and P22239-N13.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Markót, M.C., Schichl, H. Bound constrained interval global optimization in the COCONUT Environment. J Glob Optim 60, 751–776 (2014). https://doi.org/10.1007/s10898-013-0139-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-013-0139-x

Keywords

Navigation