Abstract
The convergence properties of interval global optimization algorithms are studied which select the next subinterval to be subdivided with the largest value of the indicator pf(f k ,X)=(f k −\(\underline F \)(X))/(\(\overline F \)(X)−\(\underline F \)(X)). This time the more general case is investigated, when the global minimum value is unknown, and thus its estimation f k in the iteration k has an important role. A sharp necessary and sufficient condition is given on the f k values approximating the global minimum value that ensure convergence of the optimization algorithm. The new theoretical result enables new, more efficient implementations that utilize the advantages of the pf * based interval selection rule, even for the more general case when no reliable estimation of the global minimum value is available.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
L.G. Casado and I. García, New load balancing criterion for parallel interval global optimization algorithm, in: Proc. of the 16th IASTED Internat. Conf., Garmisch-Partenkirchen, Germany, 1998, pp. 321–323.
L.G. Casado, I. García and T. Csendes, A new multisection technique in interval methods for global optimization, Computing 65 (2000) 263–269.
L.G. Casado, I. García and T. Csendes, A heuristic rejection criterion in interval global optimization BIT 41 (2001) 683–692.
L.G. Casado, I. García, T. Csendes and V.G. Ruiz, Heuristic rejection in interval global optimization, J. Optim. Theory Appl. 118 (2003) 27–43.
T. Csendes, Convergence properties of interval global optimization algorithms with a new class of interval selection criteria, J. Global Optim. 19 (2001) 307–327.
T. Csendes, Numerical experiences with a new generalized subinterval selection criterion for interval global optimization, Reliable Comput. 9 (2003) 109–125.
T. Csendes and D. Ratz, Subdivision direction selection in interval methods for global optimization, SIAM J. Numer. Anal. 34 (1997) 922–938.
R. Hammer, M. Hocks, U. Kulisch and D. Ratz, C++ Toolbox for Verified Computing (Springer, Berlin, 1995).
E. Hansen, Global Optimization Using Interval Analysis (Marcel Decker, New York, 1992).
R. Horst and P.M. Pardalos, eds., Handbook of Global Optimization (Kluwer, Dordrecht, 1995).
R.B. Kearfott, Rigorous Global Search: Continuous Problems (Kluwer, Dordrecht, 1996).
O. Knüppel, BIAS — Basic Interval Arithmetic Subroutines, Technical Report 93.3, University of Hamburg (1993).
M.Cs. Markót, T. Csendes and A.E. Csallner, Multisection in interval branch-and-bound methods for global optimization II. Numerical tests, J. Global Optim. 16 (2000) 219–228.
R.E. Moore and H. Ratschek, Inclusion functions and global optimization II, Math. Programming 41 (1988) 341–356.
H. Ratschek and J. Rokne, New Computer Methods for Global Optimization (Ellis Horwood, Chichester, 1988).
D. Ratz and T. Csendes, On the selection of subdivision directions in interval branch-and-bound methods for global optimization, J. Global Optim. 7 (1995) 183–207.
S. Skelboe, Computation of rational functions, BIT 14 (1974) 87–95.
R. Stateva and S. Tsvetkov, A diverse approach for the solution of the isothermal multiphase flash problem, application to vapor-liquid-liquid systems, Canadian J. Chem. Engrg. 72 (1994) 722–734.
A. Törn and A. Žilinskas, Global Optimization (Springer, Berlin, 1987).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Csendes, T. Generalized Subinterval Selection Criteria for Interval Global Optimization. Numerical Algorithms 37, 93–100 (2004). https://doi.org/10.1023/B:NUMA.0000049489.44154.02
Issue Date:
DOI: https://doi.org/10.1023/B:NUMA.0000049489.44154.02