Abstract
In bound constrained global optimization problems, partitioning methods utilizing Interval Arithmetic are powerful techniques that produce reliable results. Subdivision direction selection is a major component of partitioning algorithms and it plays an important role in convergence speed. Here, we propose a new subdivision direction selection scheme that uses symbolic computing in interpreting interval arithmetic operations. We call this approach symbolic interval inference approach (SIIA). SIIA targets the reduction of interval bounds of pending boxes directly by identifying the major impact variables and re-partitioning them in the next iteration. This approach speeds up the interval partitioning algorithm (IPA) because it targets the pending status of sibling boxes produced. The proposed SIIA enables multi-section of two major impact variables at a time. The efficiency of SIIA is illustrated on well-known bound constrained test functions and compared with established subdivision direction selection methods from the literature.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Berner S. (1996). New results on verified global optimization. Computing 57:323–343
Breiman L., Cutler A. (1993). A deterministic algorithm for global optimization. Math. Program. 58:179–199
Casado L.G., Martinez J.A., García I. (2001). Experiments with a new selection criterion in a fast interval optimization algorithm. J. Global Optimiz. 19:247–264
Ceberio M., Granvilliers L. (2000). Solving nonlinear systems by constraint inversion and interval arithmetic. Lecture Note Arti. Intell. 1930:127–141
Csallner A.E., Csendes T., Markót M.Cs. (2000a). Multi-section in interval branch and bound methods for global optimization I. Numerical Tests. J. Global Optimiz. 16:219–228
Csallner A.E., Csendes T., Markót M.Cs. (2000b). Multi-section in interval branch and bound methods for global optimization II Theoretical Results. J. Global Optimiz. 16:371–392
Csendes T., Klatte R., Ratz D. (2000). A Posteriori Direction Selection rules for interval optimization methods. Central Eur. J. Operation Res. 8:225–236
Csendes T., Ratz D. (1996). A review of subdivision selection in interval methods for global optimization. ZAMM Z. Angew. Math. Mech. 76:319–322
Csendes T., Ratz D. (1997). Subdivision direction selection in interval methods for global optimization. SIAM J. Numer. Anal. 34:922–938
De Jong K. (1975). An analysis of the behaviour of a class of genetic adaptive systems. PhD Thesis, University of Michigan, Ann Arbor
Granvilliers L. (2004). An interval component for continuous constraints. J. Comput. Appl. Math. 162:79–92
Granvilliers, L., Monfroy, E., Benhamou, F.: Symbolic-interval cooperation in constraint programming. Proceedings of the 2001 International Symposium on Symbolic and Algebraic Computation. London, Canada (2001)
Hammer, R., Hocks, M., Kulish, U., Ratz, D.: Numerical Toolbox for Verified computing I Berlin (1993).
Hansen E. (1992). Global optimization using interval analysis. Marcel Dekker, New York
Jansson C., Knüppel O. (1995). A branch and bound algorithm for bound constrained global optimization. J. Global Optimiz. 7:297–331
Kearfott B. (1979). An efficient degree-computation method for a generalized method of bisection. Numeric. Math. 32:109–127
Kearfott B., Kreinovich V. (1996). Applications of Interval Computations, Applied Optimization. Kluwer, Dordrecht, The Netherlands
Knüppel O. (1994). PROFIL/BIAS—A fast interval library. Computing 53:277–287
Levy A.V., Montalvo A., Gomez S., Calderon A. (1982). Topics in global optimization. Lecture Note. Math. 909:18–33
Lhomme O., Gotlieb A., Rueher M. (1998). Dynamic optimization of interval narrowing algorithms. J. Logic Program. 37:165–183
Moore R.E. (1966). Interval Analysis. Prentice-Hall, Englewood Cliffs, NJ
Moré J.J., Garbow B.S., Hillstrom K.E. (1981). Testing unconstrained optimization software. ACM Trans. Math. Software 7:17–41
Neumaier, A.: Interval Methods for Systems of Equations, Encyclopedia of Mathematics and its Applications, 37. Cambridge University Press, Cambridge (1990)
Özdamar L., Demirhan M. (2000). Experiments with new probabilistic search methods in global optimization. Comput. Operations Res. 27:841–865
Pintér J. (1992). Convergence qualification of adaptive partitioning algorithms in global optimization. Math. Program. 56:343–360
Ratschek H., Rokne J. (1995). Interval methods. In: Horst R., Pardalos P.M. (eds) Handbook of Global Optimization. Kluwer, Dordrecht, pp 751–828
Ratschek H., Rokne J. (1988). New Computer Methods for Global Optimization. John Wiley, New York
Ratz D. (1992). Automatische Ergebnisverifikation bei globalen Optimierungsproblemen. Dissertation, Universität Karlsruhe, Germany
Ratz D., Csendes T. (1995). On the selection of subdivision directions in Interval Branch-and-Bound Methods for Global Optimization. J. Global Optimiz. 7:183–207
Rosenbrock H.H. (1970). State-Space and Multivariable Theory. Wiley, New York
Sam-Haroud D., Faltings B. (1996). Consistency techniques for continuous constraints. Constraints 1:85–118
Schichl H., Neumaier A. (2005). Interval analysis on directed acyclic graphs for global optimization. J. Global Optimiz. 33:541–562
Schittkowski, K.: More Test Examples for Nonlinear Programming Codes. Lecture Notes in Economics and Mathematical Systems, vol. 282. Springer-Verlag, Berlin (1987)
Schwefel H.P. (1981). Numerical Optimization of Computer Models. Wiley, New York
Törn A., Žilinskas A. (1989). Global Optimization. Lecture Notes in Computer Science, vol. 350. Springer-Verlag, Berlin
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Pedamallu, C.S., Özdamar, L. & Csendes, T. Symbolic Interval Inference Approach for Subdivision Direction Selection in Interval Partitioning Algorithms. J Glob Optim 37, 177–194 (2007). https://doi.org/10.1007/s10898-006-9043-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10898-006-9043-y