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Experiments with a new selection criterion in a fast interval optimization algorithm

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Abstract

Usually, interval global optimization algorithms use local search methods to obtain a good upper (lower) bound of the solution. These local methods are based on point evaluations. This paper investigates a new local search method based on interval analysis information and on a new selection criterion to direct the search. When this new method is used alone, the guarantee to obtain a global solution is lost. To maintain this guarantee, the new local search method can be incorporated to a standard interval GO algorithm, not only to find a good upper bound of the solution, but also to simultaneously carry out part of the work of the interval B&B algorithm. Moreover, the new method permits improvement of the guaranteed upper bound of the solution with the memory requirements established by the user. Thus, the user can avoid the possible memory problems arising in interval GO algorithms, mainly when derivative information is not used. The chance of reaching the global solution with this algorithm may depend on the established memory limitations. The algorithm has been evaluated numerically using a wide set of test functions which includes easy and hard problems. The numerical results show that it is possible to obtain accurate solutions for all the easy functions and also for the investigated hard problems.

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References

  1. Berner, S. (1996), New Results in Verified Global Optimization. Computing 57(4), 323-343.

    Google Scholar 

  2. Casado, L.G. and García, I. (1999), Work Load Balance Approaches for Branch and Bound Algorithms on Distributed Systems. In: Proceedings of the 7th EuromicroWorkshop on Parallel and Distributed Processing. University of Madeira, Funchal, Portugal, pp. 155-162.

  3. Csendes, T. and Ratz, D. (1997), Subdivision Direction Selection in Interval Methods for Global Optimization. SIAM Journal of Numerical Analysis 34, 922-938.

    Google Scholar 

  4. Dixon, L. and Szego, G. (eds) (1975), Towards Global Optimization. Amsterdam: North-Holland Publishing Company.

    Google Scholar 

  5. Dixon, L. and Szego, G. (eds) (1978), Towards Global Optimization 2. Amsterdam: North-Holland Publishing Company.

    Google Scholar 

  6. Hammer, R., Hocks, M., Kulisch, U. and Ratz, D. (1995), C++ Toolbox for Verified Computing I: Basic Numerical Problems: Theory, Algorithms, and Programs. Springer-Verlag, Berlin.

    Google Scholar 

  7. Hansen, E. (1992), Global Optimization Using Interval Analysis, Vol. 165 of Pure and Applied Mathematics. Marcel Dekker, Inc., New York.

    Google Scholar 

  8. Henriksen, T. and Madsen, K. (1992), Use of a depth-first strategy in parallel Global Optimization. Technical Report 92-10, Institute for Numerical Analysis, Technical University of Denmark.

  9. Horst, R. and Pardalos, P. (eds) (1995), Handbook of Global Optimization, Vol. 2 of Noncovex optimization and its applications. Dordrecht, Kluwer Academic Publishers, The Netherlands.

    Google Scholar 

  10. Horst, R. and Tuy, H. (1996), Global Optimization. Deterministic Approaches (third edition). Springer-Verlag, Berlin.

    Google Scholar 

  11. Ibaraki, T. (1988), Enumerative Approaches to Combinatorial Optimisation. Annals of Operations Research 11(1-4).

  12. Ichida, K. and Fujii, Y. (1979), An Interval Arithmetic Method for Global Optimization. Computing 23: 85-97.

    Google Scholar 

  13. Knüppel, O. (1993a), BIAS-Basic Interval Arithmetic Subroutines. Technical Report 93.3, University of Hamburg.

  14. Knüppel, O. (1993b), PROFIL-Programmer's Runtime Optimized Fast Interval Library. Technical Report 93.4, University of Hamburg.

  15. Knüppel, O. and Simenec, T. (1993), PROFIL/BIAS Extensions. Technical Report 93.5, University of Hamburg.

  16. Moore, R. and Ratschek, H. (1988), Inclusion Functions and Global Optimization II. Mathematical Programming 41: 341-356.

    Google Scholar 

  17. Neumaier, A. (1999), Test functions. http://solon.mat.univie.ac.at/~vpk/math/funcs.html.

  18. Ratschek, H. and Rokne, J. (1988), New Computer Methods for Global Optimization. Ellis Horwood Ltd., Chichester.

    Google Scholar 

  19. Ratz, D. (1992), Automatische Ergebnisverifikation bei globalen Optimierungsproblemen. Ph.D. thesis, University of Karlsruhe.

  20. Ratz, D. and Csendes, T. (1995), On the Selection of Subdivision Directions in Interval Branch and Bound Methods for Global Optimization. Journal of Global Optimization 7: 183-207.

    Google Scholar 

  21. Shimano, Y., Harada, K. and Hirabayashi, R. (1997), Control schemes in a generalized utility for parallel Branch-and-Bound algorithms. In: 11th International Parallel Symposium (IPPS'97). pp. 621-627.

  22. Törn, A. and Žilinskas, A. (1989), Global Optimization, Vol. 3350. Springer-Verlag, Berlin.

    Google Scholar 

  23. Walster, G., Hansen, E. and Sengupta, S. (1985), Test results for global optimization algorithm. SIAM Numerical Optimization 1984, pp. 272-287.

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Casado, L., Martínez, J. & García, I. Experiments with a new selection criterion in a fast interval optimization algorithm. Journal of Global Optimization 19, 247–264 (2001). https://doi.org/10.1023/A:1011220023072

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