Abstract
When solving systems of nonlinear equations with interval constraint methods, it has often been observed that many calls to contracting operators do not participate actively to the reduction of the search space. Attempts to statically select a subset of efficient contracting operators fail to offer reliable performance speed-ups. By embedding the recency-weighted average Reinforcement Learning method into a constraint propagation algorithm to dynamically learn the best operators, we show that it is possible to obtain robust algorithms with reliable performances on a range of sparse problems. Using a simple heuristic to compute initial weights, we also achieve significant performance speed-ups for dense problems.
Similar content being viewed by others
References
Auer, P., Cesa-Bianchi, N., Freund, Y., & Schapire, R. E. (2002). The non-stochastic multi-armed bandit problem. SIAM Journal on Computing, 32(1), 48–77.
Auer, P., Cesa-Bianchi, N., Freund, Y., & Shapire, R. E. (1995). Gambling in a rigged casino: The adversarial multi-armed bandit problem. In Proceedings of the 36th annual symposium on foundations of computer science (FOCS ’95), pp. 322–331. IEEE Computer Society Press.
Benhamou, F. (2001). Interval constraints, interval propagation. In P. M. Pardalos, & C. A. Floudas (Eds.), Encyclopedia of Optimization, vol. 3, pp. 45–48. Kluwer Academic Publishers.
Benhamou, F., McAllester, D., & Van Hentenryck, P. (1994). CLP(Intervals) revisited. In Proceedings international symposium on logic program, pp. 124–138. The MIT Press.
Dixon, L. C. W., & Szegö, G. P. (1978). The global optimization problem: An introduction. In L. C. W. Dixon & G. P. Szegö (Eds.), Towards Global Optimization 2, pp. 1–15. North-Holland.
Duff, I. S. (1981). On algorithms for obtaining a maximum transversal. ACM Transactions on Mathematical Software, 7(3), 315–330, (September).
Goualard, F. (2005). On considering an interval constraint solving algorithm as a free-steering nonlinear gauss-seidel procedure. In Proceedings of the 20th annual ACM symposium on applied computing (reliable computation and applications track), vol. 2, pp. 1434–1438. The Association for Computing Machinery, Inc, (March).
Goualard, F., & Jermann, C. (2006). On the selection of a transversal to solve nonlinear systems with interval arithmetic. In V. N. Alexandrov et al. (Eds.), Proceedings international conference on computational science 2006, Lecture Notes in Computer Science, vol. 3991, pp. 332–339. Springer-Verlag.
Granvilliers, L., & Hains, G. (2000). A conservative scheme for parallel interval narrowing. Information Processing Letters, 74, 141–146.
Hansen. E. R., & Sengupta, S. (1981). Bounding solutions of systems of equations using interval analysis. Bibliothek Information Technologie, 21, 203–211.
Herbort, S., & Ratz, D. (1997). Improving the efficiency of a nonlinear-system-solver using a componentwise newton method. Research Report 2/1997, Institut für Angewandte Mathematik, Universität Karslruhe (TH).
Hickey, T. J., Ju, Q., & Van Emden, M. H. (2001). Interval arithmetic: From principles to implementation. J. ACM, 48(5), 1038–1068, (September).
IEEE (1990). IEEE standard for binary floating-point arithmetic. Technical Report IEEE Std 754-1985, Institute of Electrical and Electronics Engineers, 1985. Reaffirmed 1990.
INRIA Project COPRIN: Contraintes, OPtimisation, Résolution par INtervalles. The COPRIN examples page. Web page at http://www-sop.inria.fr/coprin/logiciels/ALIAS/Benches/benches.html.
Kearfott, R. B., Hu, C., & Novoa, M. III (1991). A review of preconditioners for the interval Gauss-Seidel method. Interval Computations, 1, 59–85.
Kearfott, R. B., & Shi, X. (1996). Optimal preconditioners for interval gauss-seidel methods. In G. Alefeld & A. Frommer (Eds.), Scientific Computing and Validated Numerics, pp. 173–178. Akademie Verlag.
Lebbah, Y., & Lhomme, O. (2002). Accelerating filtering techniques for numeric csps. Artificial Intelligence, 139(1), 109–132.
Luksan, L., & Vlcek, J. (1998). Sparse and partially separable test problems for unconstrained and equality constrained optimization. Research Report V767-98, Institute of Computer Science, Academy of Science of the Czech Republic, (December).
Mackworth, A. K. (1977). Consistency in networks of relations. Artificial Intelligence, 1(8), 99–118.
Moore, R. E. (1966). Interval Analysis. Prentice-Hall, Englewood Cliffs, N. J.
Moré, J. J., & Cosnard, M. Y. (1979). Numerical solutions of nonlinear equations. ACM Transactions on Mathematical Software, 5, 64–85.
Neumaier, A. (1990). Interval methods for systems of equations, Encyclopedia of Mathematics and its Applications, vol. 37. Cambridge University Press.
Ortega, J. M., & Rheinboldt, W. C. (1970). Iterative solutions of nonlinear equations in several variables. Academic Press Inc.
Ratschek, H., & Rokne, J. (1995). Interval methods. In Handbook of global optimization, pp. 751–828. Kluwer Academic.
Sotiropoulos, D. G., Nikas, J. A., & Grapsa, T. N. (2002). Improving the efficiency of a polynomial system solver via a reordering technique. In Proceedings 4th GRACM congress on computational mechanics, vol. III, pp. 970–976.
Sutton, R., & Barto, A. (1998). Reinforcement learning: An introduction. MIT Press.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Goualard, F., Jermann, C. A Reinforcement Learning Approach to Interval Constraint Propagation. Constraints 13, 206–226 (2008). https://doi.org/10.1007/s10601-007-9027-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10601-007-9027-7