Solving Nonlinear Equations by Abstraction, Gaussian Elimination, and Interval Methods

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.

[1]  Joachim Schimpf,et al.  ECLiPSe: A Platform for Constraint Logic Programming , 1997 .

[2]  Pascal Van Hentenryck,et al.  Numerica: A Modeling Language for Global Optimization , 1997, IJCAI.

[3]  John W. Lloyd.  Beyond the Glass Box: Constraints as Objects , 1995 .

[4]  Roland H. C. Yap,et al.  The CLP( R ) language and system , 1992, TOPL.

[5]  Michel Rueher,et al.  A Distributed Cooperating Constraints Solving System , 1995, Int. J. Artif. Intell. Tools.

[6]  Frédéric Goualard,et al.  An Extension of the WAM for Hybrid Interval Solvers , 1999, J. Funct. Log. Program..

[7]  Eric Monfroy,et al.  Symbolic-interval cooperation in constraint programming , 2001, ISSAC '01.

[8]  Frédéric Benhamou,et al.  Applying Interval Arithmetic to Real, Integer, and Boolean Constraints , 1997, J. Log. Program..

[9]  M. V. van Emden,et al.  Algorithmic Power from Declarative Use of Redundant Constraints , 1999 .

[10]  K. Yamamura,et al.  Interval solution of nonlinear equations using linear programming , 1998 .

[11]  Kiyotaka Yamamura.  Interval Analysis Using Linear Programming , 1997 .

[12]  M. H. van Emden,et al.  Algorithmic Power from Declarative Use of Redundant Constraints , 1999, Constraints.

[13]  David A. McAllester,et al.  Solving Polynomial Systems Using a Branch and Prune Approach , 1997 .

[14]  Alain Colmerauer,et al.  Constraint logic programming: selected research , 1993 .

[15]  Martine Ceberio,et al.  Solving Nonlinear Systems by Constraint Inversion and Interval Arithmetic , 2000, AISC.

[16]  Alain Colmerauer.  Naive Solving of Non-linear Constraints , 1991, WCLP.

[17]  Timothy J. Hickey.  CLIP: A CLP(Intervals) Dialect for Metalevel Constraint Solving , 2000, PADL.

[18]  Pascal Van Hentenryck,et al.  CLP(Intervals) Revisited , 1994, ILPS.

[19]  Laurent Granvilliers.  A Symbolic-Numerical Branch and Prune Algorithm for Solving Non-linear Polynomial Systems , 1998, J. Univers. Comput. Sci..