Solving Nonlinear Equations by Abstraction, Gaussian Elimination, and Interval Methods
- 286 Downloads
The solving engines of most of constraint programming systems use interval-based consistency techniques to process nonlinear systems over the reals. However, few symbolic-interval cooperative solvers are implemented. The challenge is twofold: control of the amount of symbolic computations, and prediction of the accuracy of interval computations over transformed systems.
In this paper, we introduce a new symbolic pre-processing for interval branch-and-prune algorithms based on box consistency. The symbolic algorithm computes a linear relaxation by abstraction of the nonlinear terms. The resulting rectangular linear system is processed by Gaussian elimination. Control strategies of the densification of systems during elimination are devised. Three scalable problems known to be hard for box consistency are efficiently solved.
KeywordsSymbolic Computation Gaussian Elimination Interval Arithmetic Linear Relaxation Real Interval
Unable to display preview. Download preview PDF.
- 2.F. Benhamou, D. McAllester, and P. Van Hentenryck. CLP(Intervals) Revisited.In Procs. of ILPS'94, Intl. Logic Prog. Symp., pages 124–138, Ithaca, USA, 1994. MIT Press.Google Scholar
- 4.M. Ceberio and L. Granvilliers. Solving Nonlinear Systems by Constraint Inversion and Interval Arithmetic. In Procs. of AISC’2000, 5th Intl. Conf. on Artificial Intelligence and Symbolic Computation, volume 1930 of LNAI, Madrid, Spain, 2000. Springer-Verlag.Google Scholar
- 5.A. Colmerauer. Naive Solving of Non-linear Constraints. In F. Benhamou and A. Colmerauer, eds., Constraint Logic Programming: Selected Research, pages 89–112. MIT Press, 1993.Google Scholar
- 6.F. Goualard, F. Benhamou, and L. Granvilliers. An Extension of the WAM for Hybrid Interval Solvers. J. of Functional and Logic Programming, 5(4):1–31, 1999.Google Scholar
- 8.L. Granvilliers, E. Monfroy, and F. Benhamou. Symbolic-Interval Cooperation in Constraint Programming. In Procs. of ISSAC’2001, 26th Intl. Symp. on Symbolic and Algebraic Computation, pages 150–166, Univ. of Western Ontario, London, Ontario, Canada, 2001. ACM Press.Google Scholar
- 9.T. J. Hickey. CLIP: a CLP(Intervals) Dialect for Metalevel Constraint Solving. In Procs. of PADL’2000, Intl. Workshop on Practical Aspects of Declarative Languages, volume 1753 of LNCS, pages 200–214, Boston, USA, 2000. Springer-Verlag.Google Scholar
- 10.H. Hong. RISC-CLP(Real): Constraint Logic Programming over Real Numbers. In F. Benhamou and A. Colmerauer, eds., Constraint Logic Programming: Selected Research. MIT Press, 1993.Google Scholar
- 14.J.-F. Puget and M. Leconte. Beyond the Glass Box: Constraints as Objects. In Procs. of ILPS’95, Intl. Logic Programming Symposium, pages 513–527, Portland, USA, 1995. MIT Press.Google Scholar
- 15.A. Semenov and A. Leshchenko. Interval and Symbolic Computations in the Unicalc Solver. In Procs. of INTERVAL’94, pages 206–208, St-Petersburg, Russia, 1994.Google Scholar
- 18.P. Van Hentenryck, L. Michel, and Y. Deville. Numerica: a Modeling Language for Global Optimization. MIT Press, 1997.Google Scholar
- 19.M. Wallace, S. Novello, and J. Schimpf. ECLiPSe: A Platform for Constraint Logic Programming. Technical report, IC-Parc, London, 1997.Google Scholar