Efficient and safe global constraints for handling numerical constraint systems

SIAM Journal on Numerical Analysis

Volumn 42, Issue 5, 2005, Pages 2076-2097

  Lebbah, Yahia ^a,d   Michel, Claude ^b,d   Rueher, Michel ^b,d   Daney, David ^c,d   Merlet, Jean Pierre ^c,d  

^a UNIVERSITY OF ORAN   (Algeria)

^b UNIVERSITÉ CÔTE D'AZUR   (France)

^c INRIA   (France)

^d UNIVERSITY OF NICE SOPHIA ANTIPOLIS   (France)

Author keywords

Constraint programming; Global constraint; Interval arithmetic; Reformulation linearization technique; Safe rounding; Systems of equations and inequalities

Indexed keywords

ALGORITHMS; BENCHMARKING; LINEAR SYSTEMS; OPTIMIZATION; POLYNOMIALS; PROBLEM SOLVING;

CONSTRAINT PROGRAMMING; GLOBAL CONSTRAINT; INTERVAL ARITHMETIC; REFORMULATION LINEARIZATION TECHNIQUE; SAFE ROUNDING; SYSTEMS OF EQUATIONS AND INEQUALITIES;

CONSTRAINT THEORY;

EID: 14644426280     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036142903436174     Document Type: Article

References (63)

1. F. A. AL-KHAYYAL AND J. E. FALK, Jointly constrained biconvex programming, Math. Oper. Res., 8 (1983), pp. 273-286.
2. E. ALLGOWER AND K. GEORG, Numerical Continuation Methods: An Introduction, Springer-Verlag, Berlin, 1990.
3. C. AUDET, P. HANSEN, B. JAUMARD, AND G. SAVARD, Branch and cut algorithm for nonconvex quadratically constrained quadratic programming, Math. Program., 87 (2000), pp. 131-152.
4. A. B. BABICHEV, O. P. KADYROVA, T. P. KASHEVAROVA, A. S. LESHCHENKO, AND A. L. SEMENOV, Unicalc, a novel approach to solving systems of algebraic equations, Interval Computations, 2 (1993), pp. 29-47.
5. F. BENHAMOU, F. GOUALARD, L. GRANVILLIERS, AND J.-F. PUGET, Revising hull and box consistency, in Proceedings of ICLP '99, 1999, MIT Press, pp. 230-244.
6. F. BENHAMOU, D. MCALLESTER, AND P. VAN-HENTENRYCK, CLP (intervals) revisited, in Proceedings of the International Symposium on Logic Programming, MIT Press, Cambridge, MA, 1994, pp. 124-138.
7. F. BENHAMOU AND W. OLDER, Applying interval arithmetic to real, integer and Boolean constraints, J. Logic Programming, 32 (1997), pp. 1-24.
8. B. BOTELLA AND P. TAILLIBERT, Interlog: Constraint logic programming on numeric intervals, in 3rd International Workshop on Software Engineering, Artificial Intelligence and Expert Systems, Oberammergau, 1993.
9. R. L. BURDEN AND J. D. FAIRES, Numerical Analysis, 5th ed., PWS-KENT, Boston, MA, 1993.
10. J. C. CLEARY, Logical arithmetic, Future Computing Systems, 2 (1987), pp. 125-149.
11. H. COLLAVIZZA, F. DELOBEL, AND M. RUEHER, Comparing partial consistencies, Reliable Computing, 5 (1999), pp. 213-228.
12. A. COLMERAUER, Spécifications de Prolog IV, Technical report, GIA, Faculté des Sciences de Luminy, France, 1994.
13. D. Cox, J. LITTLE, AND D. O'SHEA, Ideals, Varieties, and Algorithms, 2nd ed., Springer-Verlag, New York, 1997.
14. Y. CRAMA, Recognition problems for polynomial in 0-1 variables, Math. Progr., 44 (1989), pp. 139-155.
15. O. DIDRIT, Analyse par Intervalles pour L'automatique: Résolution Globale et Garantie de Problèmes Non Linéaires en Robotique et en Commande Robuste, Ph.D. thesis, Université Parix XI Orsay, France, 1997.
16. P. DIETMAIER, The Stewart-Gough platform of general geometry can have 40 real postures, in Advances in Robot Kinematics: Analysis and Control, Kluwer, Dordrecht, 1998, pp. 1-10.
17. C. A. FLOUDAS, ED., Deterministic Global Optimization: Theory, Algorithms and Applications, Kluwer, Dordrecht, 2000.
18. D. GOLDBERG, What every computer scientist should know about floating-point arithmetic, ACM Computing Surveys, 23 (1991), pp. 5-48.
19. E. HANSEN AND S. SENGUPTA, Bounding solutions of systems of equations using interval analysis, BIT, 21 (1981), pp. 203-221.
20. E. R. HANSEN, Global Optimization Using Interval Analysis, Marcel Dekker, New York, 1992.
21. H. HONG AND V. STAHL, Starting regions by fixed points and tightening, Computing, 53 (1994), pp. 323-335.
22. ILOG Solver 4.0, Reference Manual, ILOG, Mountain View, 1997.
23. R. B. KEARFOTT, Test of generalized bisection, ACM Trans. Math. Software, 13 (1987), pp. 197-220.
24. R. KRAWCZYK, Newton-Algorithmen zur Bestimmung von Nullstellen mit Fehlerschranken, Computing, 4 (1969), pp. 187-201.
25. V. KREINOVICH, A. LAKEYEV, J. ROHN, AND P. KAHL, Computational Complexity and Feasibility of Data Processing and Interval Computations, Kluwer, Dordrecht, 1998.
26. Y. LEBBAH AND O. LHOMME, Accelerating filtering techniques for numeric CSPs, Artificial Intelligence, 139 (2002), pp. 109-132.
27. Y. LEBBAH, M. RUEHER, AND C. MICHEL, A global filtering algorithm for handling systems of quadratic equations and inequations, in Proc. of the 8th International Conference on Principles and Practice of Constraint Programming, Cornell University, New York, 2002, Lecture Notes in Comput. Sci. 2470, pp. 109-123.
28. J. H. M. LEE AND M. H. VAN EMDEN, Interval computation as deduction in CHIP, J. Logic Programming, 16 (1993), pp. 255-276.
29. T.-Y. LEE AND J.-K. SHIM, Elimination-based solution method for the forward kinematics of the general Stewart-Gough platform, in Computational Kinematics, F. C. Park and C. C. Iurascu, eds., 2001, pp. 259-267.
30. O. LHOMME, Consistency techniques for numeric CSPs, in Proceedings of International JointConference on Artificial Intelligence, Chambéry, France, 1993, pp. 232-238.
31. A. MACKWORTH, Consistency in networks of relations, J. Artificial Intelligence, 8 (1977), pp. 99-118.
32. G. P. MCCORMICK, Computability of global solutions to factorable nonconvex programs - Part I - Convex underestimating problems, Math. Progr., 10 (1976), pp. 147-175.
33. C. MICHEL, Y. LEBBAH, AND M. RUEHER, Safe embedding of the simplex algorithm in a CSP framework, in Proc. of 5th Int. Workshop on Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization Problems CPAIOR 2003, CRT, Université de Montréal, 2003, pp. 210-220.
34. U. MONTANARI, Networks of constraints: Fundamental properties and applications to image processing, Inform. Sci., 7 (1974), pp. 95-132.
35. R. MOORE, Interval Analysis, Prentice-Hall, Englewood Cliff, NJ, 1966.
36. R. E. MOORE, Methods and Applications of Interval Analysis, SIAM Stud. Appl. Math. 2, SIAM, Philadelphia, 1979.
37. A. P. MORGAN, Computing all solutions to polynomial systems using homotopy continuation, Appl. Math. Comput., 24 (1987), pp. 115-138.
38. A. NEUMAIKH, Interval Methods for Systems of Equations, Encyclopedia Math. Appl. 37, Cambridge University Press, Cambridge, 1990.
39. A. NEUMAIER, The wrapping effect, ellipsoid arithmetic, stability and confidence region, Comput, Suppl., 9 (1993). pp. 175-190.
40. A. NEUMAIER, Introduction to Numerical Analysis, Cambridge University Press, Cambridge, 2001.
41. A. NEUMAIER, Complete, search in continuous global optimization and constraint satisfaction, in Acta Nurnerica 2004, A. Iserles. ed., Cambridge University Press, Cambridge, UK, 2004, pp. 271-369.
42. A. NEUMAIER AND O. SHCHERRINA, Safe bounds in linear and mixed-integer programming, Math. Progr. A, 99 (2004), pp. 283-296.
43. W. J. OLDER AND A. VELINO, Extending prolog with constraint arithmetic on real intervals, in Proc. of IEEE Canadian Conference on Electrical and Computer Engineering, IEEE Computer Society Press, 1990, pp. 14.1.1-14.1.4.
44. G. PESANT AND M. BOYER, QUAD-CLP(R): Adding the power of quadratic constraints, in Principles and Practice of Constraint Programming '94, Lecture Notes in Comput. Sci. 874, 1994, pp. 95-107.
45. G. PESANT AND M. BOYER, Reasoning about solids using constraint logic programming, J. Automat. Reason., 22 (1999), pp. 241-262.
46. H. RATSCHEK AND J. ROKNE, Computer Methods for the Range of Functions, Ellis Horwood Ser. Math. Appl., Ellis Horwood, New York, 1984.
47. N. REVOL AND F. ROUILLIER, Motivations for an arbitrary precision interval arithmetic and the MPFI library, in Workshop on Validated Computing, Toronto, Canada, 2002.
48. A. RIKUN, A convex envelope formula for multilinear functions, J. Global Optim., 10 (1997), pp. 425-437.
49. H. S. RYOO AND N. V. SAHINIDIS, Analysis of bounds for multilinear functions, J. Global Optim., 19 (2001), pp. 403-424.
50. N. V. SAHINIDIS AND M. TWARMALANI, BARON 5.0: Global Optimization of Mixed-Integer Nonlinear Programs, Technical report, Department of Chemical and Biomolecular Engineering, University of Illinois at Urbana-Champaign, 2002.
51. N. V. SAHINIDIS AND M. TWARMALANI, Global optimization of mixed-integer programs: A theoretical and computational study, Math. Progr., 99 (2004), pp. 563-591.
52. H. D. SHERALI, Convex envelopes of multilinear functions over a unit hypercube and over special discrete sets, Acta Math. Vietnam., 22 (1997), pp. 245-270.
53. H. D. SHERALI AND W. P. ADAMS, A Reformulation-Linearization Technique for Solving Discrete and Continuous Nonconvex Problems, Kluwer, Dordrecht, 1999.
54. H. D. SHERALI AND C. H. TUNCBILEK, A global optimization algorithm for polynomial using a reformulation-linearization technique, J. Global Optim., 2 (1992), pp. 101-112.
55. H. D. SHERALI AND C. H. TUNCBILEK, A reformulation-convexification approach for solving nonconvex quadratic programming problems, J. Global Optim., 7 (1995), pp. 1-31.
56. H. D. SHERALI AND C. H. TUNCBILEK, A comparison of two reformulation-linearization technique based on linear programming relaxations for polynomial porgramming problems, J. Global Optim., 10 (1997), pp. 381-390.
57. H. D. SHERALI AND C. H. TUNCBILEK, New reformulation linearization/ convexification relaxations for univariate and multivariate polynomial programming problems, Oper. Res. Lett., 21 (1997), pp. 1-9.
58. N. Z. SHOR, Dual quadratic estimates in polynomial and Boolean programming, Ann. Oper. Res., 25 (1990), pp. 163-168.
59. M. TAWARMALANI AND N. V. SAHINIDIS, EDS., Convexification and Global Optimization in Continuous and Mixed-Integer Non-Linear Programming, Kluwer, Dordrecht, 2002.
60. P. VAN HENTENRYCK, D. MCALLESTER, AND D. KAPUR, Solving polynomial systems using a branch and prune approach, SIAM J. Numer. Anal., 34 (1997), pp. 797-827.
61. P. VAN-HENTENRYCK, L. MICHEL, AND Y. DEVILLE, Numerica: A Modeling Language for Global Optimization, MIT Press, Cambridge, MA, 1997.
62. J. VERSCHELDE, Algorithm 795: PHCPACK: A general-purpose solver for polynomial systems by homotopy continuation, ACM Trans. Math. Software, 25 (1999), pp. 251-276.
63. K. YAMAMURA, H. KAWATA, AND A. TOKUE, Interval solution of nonlinear equations using linear programming, BIT, 38 (1998), pp. 186-199.