H. Batnini, C. Michel, and M. Rueher, Une nouvelle stratégie de recherche pour les csps continus, Premières Journées Francophones de Programmation par Contraintes (JFPC'05), Lens - Université d'Artois, pp.8-10

S. Mokhtar, . Bazaraa, D. Hanif, . Sherali-et-marakada-chitharanjan, and . Shetty, Nonlinear Programming : Theory and Algorithms, 1993.

M. Belaid, Résolution de contraintes sur les flottants dédiée à la vérification de programmes, Thèse de doctorat, 2013.

M. Said-belaid, C. Michel, and M. Rueher, Approximating floating-point operations to verify numerical programs, 14th GAMM-IMACS International Symposium on Scientific Computing , Computer Arithmetic and Validated Numerics (SCAN'10), pp.27-30

M. Said-belaid, C. Michel, and M. Rueher, Boosting Local Consistency Algorithms over Floating-Point Numbers, Proceedings of the 18th international conference on Principles and practice of constraint programming (CP'12), pp.127-140
DOI : 10.1007/978-3-642-33558-7_12

F. Benhamou and F. Goualard, Laurent Granvilliers et Jean-François Puget : Revising hull and box consistency, INT

F. Benhamou, F. Goualard, É. Languénou, and M. Christie, Interval constraint solving for camera control and motion planning, ACM Transactions on Computational Logic, vol.5, issue.4, pp.732-767, 2004.
DOI : 10.1145/1024922.1024927

F. Benhamou, D. Mcallester, and P. Van-hentenryck, Clp(intervals) revisited, Proceedings of the International Symposium on Logic Programming, pp.124-138, 1994.

S. Boldo and J. Filliâtre, Formal Verification of Floating-Point Programs, 18th IEEE Symposium on Computer Arithmetic (ARITH '07), pp.187-194
DOI : 10.1109/ARITH.2007.20
URL : https://hal.archives-ouvertes.fr/hal-01174892

G. Borradaile and P. Van-hentenryck, Safe and tight linear estimators for global optimization, Mathematical Programming, pp.495-517, 2005.
DOI : 10.1007/s10107-004-0533-8

B. Botella, A. Gotlieb, and C. Michel, Symbolic execution of floating-point computations. Software Testing, Verification and Reliability, pp.97-121, 2006.
URL : https://hal.archives-ouvertes.fr/inria-00540299

B. Botella, A. Gotlieb, C. Michel, M. Rueher, and P. Taillibert, Utilisation des contraintes pour la génération automatique de cas de test structurels, Techniques et science informatiques (TSI), pp.1163-1187, 2002.

A. Brillout, D. Kroening, and T. Wahl, Mixed abstractions for floating-point arithmetic, 2009 Formal Methods in Computer-Aided Design, pp.69-76
DOI : 10.1109/FMCAD.2009.5351141

L. Chen, A. Miné, and P. Cousot, A sound floatingpoint polyhedra abstract domain, Proc. of the Sixth Asian Symposium on Programming Languages and Systems (APLAS'08), volume 5356 de Lecture Notes in Computer Science (LNCS), pp.3-18, 2008.
URL : https://hal.archives-ouvertes.fr/hal-00531567

E. Clarke, D. Kroening, and F. Lerda, A Tool for Checking ANSI-C Programs, TACAS, volume 2988 de LNCS, pp.168-176, 2004.
DOI : 10.1007/978-3-540-24730-2_15

H. Collavizza, F. Delobel, and M. Rueher, A note on partial consistencies over continuous domains solving techniques, Proceedings of CP98 (Fourth International Conference on Principles and Practice of Constraint Programming), pp.26-30, 1998.

H. Collavizza, F. Delobel, and M. Rueher, Extending consistent domains of numeric csp, Proceedings of the 16th International Joint Conference on Artifical Intelligence -Volume 1, IJCAI'99, pp.406-411, 1999.

H. Collavizza, C. Michel, O. Ponsini, and M. Rueher, Generating test cases inside suspicious intervals for floating-point number programs, Proceedings of the 6th International Workshop on Constraints in Software Testing, Verification, and Analysis, CSTVA 2014, pp.31-2014
DOI : 10.1145/2593735.2593737
URL : https://hal.archives-ouvertes.fr/hal-01099503

H. Collavizza, M. Rueher, and P. Van-hentenryck, CPBPV: a constraint-programming framework for bounded program verification, Constraints, vol.404, issue.3, pp.238-264, 2010.
DOI : 10.1007/s10601-009-9089-9
URL : https://hal.archives-ouvertes.fr/hal-01099509

R. D. , C. , O. Lhomme, and C. Michel, Aligning component upgrades, Second Workshop on Logics for Component Configuration, pp.1-11
URL : https://hal.archives-ouvertes.fr/hal-00622830

P. Cousot and R. Cousot, Abstract interpretation, Proceedings of the 4th ACM SIGACT-SIGPLAN symposium on Principles of programming languages , POPL '77, pp.238-252, 1977.
DOI : 10.1145/512950.512973
URL : https://hal.archives-ouvertes.fr/hal-01108790

P. Cousot, R. Cousot, J. Feret, and L. Mauborgne, Antoine Miné et Xavier Rival : Why does astrée scale up ? Formal Methods in System Design, pp.229-264, 2009.

D. Delmas, E. Goubault, S. Putot, and J. Souyris, Karim Tekkal et Franck Védrine : Towards an industrial use of FLUC- TUAT on safety-critical avionics software, FMICS, pp.53-69, 2009.

G. Delzanno and A. Podelski, Model checking in CLP In Tools and Algorithms for the Construction and Analysis of Systems, 5th International Conference, TACAS '99, pp.22-28

T. Denmat, A. Gotlieb, and M. Ducassé, An Abstract Interpretation Based Combinator for Modelling While Loops in Constraint Programming, 13th International Conference on Principles and Practice of Constraint Programming, pp.241-255
DOI : 10.1007/978-3-540-74970-7_19
URL : https://hal.archives-ouvertes.fr/hal-00180548

P. Dietmaier, The Stewart-Gough Platform of General Geometry can have 40 Real Postures, Advances in Robot Kinematics : Analysis and Control, pp.1-10, 1998.
DOI : 10.1007/978-94-015-9064-8_1

F. Domes-et-arnold-neumaier-kun-li, E. B. Kosmatopoulos, P. A. Ioannou, and H. , Rigorous filtering using linear relaxations Ryaciotaki-Boussalis : Quantified multivariate polynomial inequalities . the mathematics of practical control design problems, Journal of Global Optimization IEEE Control Systems Magazine, vol.53, issue.35, pp.441-47348, 2000.

P. Dorato, W. Yang, and C. Abdallah, Robust Multi-Objective Feedback Design by Quantifier Elimination, Journal of Symbolic Computation, vol.24, issue.2, pp.153-159, 1997.
DOI : 10.1006/jsco.1997.0120

D. Vijay, L. Silva, D. Haller, M. Kroening, and . Tautschnig, Numeric bounds analysis with conflict-driven learning, Proc. TACAS, pp.48-63, 2012.

G. Fiorio, S. A. Malan, M. Milanese, and M. Taragna, Robust performance design of fixed structure controllers for systems with uncertain parameters, Proceedings of 32nd IEEE Conference on Decision and Control, pp.3029-3031
DOI : 10.1109/CDC.1993.325757

A. Frommer and B. Lang, Existence Tests for Solutions of Nonlinear Equations Using Borsuk's Theorem, SIAM Journal on Numerical Analysis, vol.43, issue.3, pp.1348-1361, 2006.
DOI : 10.1137/S0036142903438148

A. Goldsztejn, A branch and prune algorithm for the approximation of non-linear AE-solution sets, Proceedings of the 2006 ACM symposium on Applied computing , SAC '06, pp.1650-1654, 2006.
DOI : 10.1145/1141277.1141665
URL : https://hal.archives-ouvertes.fr/hal-00481391

A. Goldsztejn and L. Jaulin, Inner and Outer Approximations of Existentially Quantified Equality Constraints, Proceedings of the Twelfth International Conference on Principles and Practice of Constraint Programming, pp.198-212, 2006.
DOI : 10.1007/11889205_16
URL : https://hal.archives-ouvertes.fr/hal-00481288

A. Goldsztejn, Y. Lebbah, C. Michel, and M. Rueher, Revisiting the Upper Bounding Process in a Safe Branch and Bound Algorithm, Proceedings of the 14th international conference on Principles and Practice of Constraint Programming (CP'08), pp.598-602, 2008.
DOI : 10.1007/978-3-540-85958-1_49
URL : https://hal.archives-ouvertes.fr/hal-00297086

A. Goldsztejn, C. Michel, and M. Rueher, An efficient algorithm for a sharp approximation of universally quantified inequalities, Proceedings of the 2008 ACM symposium on Applied computing , SAC '08, pp.134-139, 2008.
DOI : 10.1145/1363686.1363724
URL : https://hal.archives-ouvertes.fr/inria-00293749

A. Goldsztejn, C. Michel, and M. Rueher, Efficient handling of universally quantified inequalities, Constraints, vol.43, issue.3, pp.117-135, 2009.
DOI : 10.1007/s10601-008-9053-0
URL : https://hal.archives-ouvertes.fr/hal-00431129

A. Gotlieb, B. Botella, and M. Rueher, Automatic Test Data Generation Using Constraint Solving Techniques, Proceedings of ISSTA'98, pp.53-62, 1998.

A. Gotlieb, B. Botella, and M. Rueher, A CLP Framework for Computing Structural Test Data, Computational Logic -CL 2000, First International Conference Proceedings, pp.24-28, 2000.
DOI : 10.1007/3-540-44957-4_27

E. Goubault, Static Analyses of the Precision of Floating-Point Operations, Static Analysis, 8th International Symposium, pp.234-259, 2001.
DOI : 10.1007/3-540-47764-0_14

E. Goubault and S. Putot, Static Analysis of Numerical Algorithms, SAS, pp.18-34, 2006.
DOI : 10.1007/11823230_3

E. Goubault and S. Putot, Robustness Analysis of Finite Precision Implementations, Programming Languages and Systems - 11th Asian Symposium, APLAS, volume 8301 de LNCS, pp.50-57, 2013.
DOI : 10.1007/978-3-319-03542-0_4

L. Haller, A. Griggio, M. Brain, and D. Kroening, Deciding floating-point logic with systematic abstraction, Proceedings of Formal Methods in Computer-Aided Design (FMCAD) 2012, pp.131-140

E. Hansen and R. Greenberg, An interval Newton method, Applied Mathematics and Computation, vol.12, issue.2-3, pp.1289-98, 1983.
DOI : 10.1016/0096-3003(83)90001-2

E. Hansen and G. W. Walster, Global Optimization Using Interval Analysis : Revised and Expanded, 2003.

E. Hansen and G. W. Walster, AN OVERVIEW OF GLOBAL OPTIMIZATION USING INTERVAL ANALYSIS, 2004.
DOI : 10.1016/B978-0-12-505630-4.50021-3

T. J. Hickey, Q. Ju-et-maarten, and H. Van-emden, Interval arithmetic: From principles to implementation, Journal of the ACM, vol.48, issue.5, pp.1038-1068, 2001.
DOI : 10.1145/502102.502106

S. Hongthong, R. Baker, and . Kearfott, Rigorous linear overestimators and underestimators. Rapport technique, university of Lousiana, 2004.

R. Horst and H. Tuy, Global Optimization : Deterministic Approches, 1993.

L. Jaulin and É. Walter, Guaranteed tuning, with application to robust control and motion planning, Automatica, vol.32, issue.8, pp.1217-1221, 1996.
DOI : 10.1016/0005-1098(96)00050-7

M. Jirstrand, R. Baker, and . Kearfott, Nonlinear control system design by quantifier elimination Journal of Symbolic Computation -Special issue : applications of quantifier elimination archive, pp.137-152, 1996.

R. Baker and . Kearfott, Validated probing with linear relaxations. submitted to Journal of Global Optimization, 2005.

R. Baker and . Kearfott, Discussion and empirical comparisons of linear relaxations and alternate techniques in validated deterministic global optimization. journal of Optimization Methods and Software, pp.715-731, 2006.

Y. Lebbah, C. Michel, and M. Rueher, Efficient Pruning Technique Based on Linear Relaxations, Second international conference on Global Optimization and Constraint Satisfaction (CO- COS'03), pp.18-21
DOI : 10.1007/11425076_1

Y. Lebbah, C. Michel, and M. Rueher, An efficient and safe framework for solving optimisation problems, 11th GAMM - IMACS International Symposium on Scientific Computing, Computer Arithmetic, and Validated Numerics, pp.4-8

Y. Lebbah, C. Michel, and M. Rueher, Efficient Pruning Technique Based on Linear Relaxations, Global Optimization and Constraint Satisfaction (selected papers from COCOS'03), pp.1-14, 2005.
DOI : 10.1007/11425076_1

Y. Lebbah, C. Michel, and M. Rueher, A Rigorous Global Filtering Algorithm for Quadratic Constraints*, Constraints, vol.7, issue.1, pp.47-65, 2005.
DOI : 10.1007/s10601-004-5307-7

Y. Lebbah, C. Michel, and M. Rueher, An efficient and safe framework for solving optimization problems, Journal of Computational and Applied Mathematics, vol.199, issue.2, pp.372-377, 2007.
DOI : 10.1016/j.cam.2005.08.037
URL : https://hal.archives-ouvertes.fr/hal-00510304

Y. Lebbah, C. Michel, and M. Rueher, Using constraint techniques for a safe and fast implementation of optimality-based reduction, Proceedings of the 2007 ACM symposium on Applied computing , SAC '07, pp.326-331, 2007.
DOI : 10.1145/1244002.1244079

Y. Lebbah, C. Michel, and M. Rueher, Utilisation des techniques de programmation par contraintes pour une implémentation rigoureuse et efficace de la réduction basée sur l'optimalité, Troisièmes Journées Francophones de Programmation par Contraintes, pp.4-6, 2007.

Y. Lebbah, C. Michel, M. Rueher, D. Daney, and J. Merlet, Efficient and Safe Global Constraints for Handling Numerical Constraint Systems, SIAM Journal on Numerical Analysis, vol.42, issue.5, pp.2076-2097, 2005.
DOI : 10.1137/S0036142903436174
URL : https://hal.archives-ouvertes.fr/hal-00907765

Y. Lebbah, M. Rueher, and C. Michel, A Global Filtering Algorithm for Handling Systems of Quadratic Equations and Inequations, Proceedings of the 8th International Conference on Principles and Practice of Constraint Programming (CP'02), pp.109-123
DOI : 10.1007/3-540-46135-3_8

O. Lhomme, Consistency techniques for numeric csps, Proceedings of IJCAI'93, pp.232-238, 1993.

L. Liberti and C. Constantinos, Pantelides : Convex envelopes of monomials of odd degree, Journal of Global Optimization, vol.25, issue.2, pp.157-168, 2003.
DOI : 10.1023/A:1021924706467

A. K. Mackworth, Consistency in networks of relations, Artificial Intelligence, vol.8, issue.1, pp.99-118, 1977.
DOI : 10.1016/0004-3702(77)90007-8

S. A. Malan, M. Milanese, and M. Taragna, Robust Analysis and Design of Control Systems Using Interval Arithmetic**A previous version of this paper was presented at the 13th IFAC World Congress, which was held in San Francisco, CA, during 30 June-5 July, 1996. The Published Proceedings of this IFAC Meeting may be ordered from: Elsevier Science Limited, The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, U.K. This paper was recommended for publication in revised form by Associate Editor F. Callier under the direction of Editor R.F. Curtain., Automatica, vol.33, issue.7, pp.1363-1372, 1997.
DOI : 10.1016/S0005-1098(97)00028-9

B. Marre and C. Michel, Improving the Floating Point Addition and Subtraction Constraints, Proceedings of the 16th international conference on Principles and practice of constraint programming (CP'10), pp.360-367
DOI : 10.1007/978-3-642-15396-9_30

G. P. Mccormick, Computability of global solutions to factorable nonconvex programs: Part I ??? Convex underestimating problems, Mathematical Programming, pp.147-175, 1976.
DOI : 10.1287/mnsc.17.11.759

C. Michel, Modèles et Implémentations d'Interprètes Réflexifs, Thèse de doctorat, 1997.

C. Michel, Exact projection functions for floating point number constraints, Seventh international symposium on Artificial Intelligence and Mathematics (7th ISAIM), Fort Lauderdale, Floride (US), pp.2-4, 2002.

C. Michel, M. S. Belaid, O. Ponsini, and M. Rueher, An hybrid approach to solve constraints over the floatingpoint numbers, 13th INFORMS Computing Society Conference (ICS), pp.6-8

C. Michel, Y. Lebbah, and M. Rueher, Safe embedding of the simplex algorithm in a csp framework, 5th Int. Workshop on Integration of AI and OR techniques in Constraint Programming for Combinatorial Optimisation Problems (CPAIOR'03), pp.210-220, 2003.

C. Michel and M. Rueher, Handling software upgradeability problems with milp solvers Claude Michel, Michel Rueher et Yahia Lebbah : Solving constraints over floating-point numbers, First Workshop on Logics for Component Configuration (LoCoCo'10), volume 29 de EPTCS Proceedings of the 7th International Conference on Principles and Practice of Constraint Programming (CP'01), LNCS 2239, pp.1-10, 2001.
DOI : 10.4204/eptcs.29.1
URL : http://doi.org/10.4204/eptcs.29.1

A. Miné, Weakly relational numerical abstract domains, Thèse de doctorat, École Polytechnique, 2004.

M. Minoux, Mathematical Programming. Theory, Algorithms and Applications, 1986.

A. Miné, Relational Abstract Domains for the Detection of Floating-Point Run-Time Errors, Proc. of the European Symposium on Programming volume 2986 de Lecture Notes in Computer Science (LNCS), pp.3-17, 2004.
DOI : 10.1007/978-3-540-24725-8_2

A. Bruce, . Murtagh, and A. Michael, Saunders : Minos 5.5 user's guide, 1998.

A. Neumaier, Complete search in continuous global optimization and constraint satisfaction, Acta Numerica, 2004.

A. Neumaier and O. Shcherbina, Safe bounds in linear and mixed-integer linear programming, Mathematical Programming, pp.283-296, 2004.
DOI : 10.1007/s10107-003-0433-3

B. Neveu, G. Trombettoni, and I. Araya, Node selection strategies in interval Branch and Bound algorithms, Journal of Global Optimization, vol.103, issue.2, pp.289-304, 2016.
DOI : 10.1007/s10898-015-0375-3
URL : https://hal.archives-ouvertes.fr/hal-01230893

M. James, . Ortega, and C. Werner, Rheinboldt : Iterative Solution of Nonlinear Equations in Several Variables, 1970.

M. Pelleau, C. Truchet, and F. Benhamou, The octagon abstract domain for continuous constraints, Constraints, vol.19, issue.1, pp.309-337, 2014.
DOI : 10.1007/s10601-014-9162-x
URL : https://hal.archives-ouvertes.fr/hal-01147912

A. Podelski, Model Checking as Constraint Solving, Static Analysis, 7th International Symposium, SAS 2000, pp.22-37, 2000.
DOI : 10.1007/978-3-540-45099-3_2

O. Ponsini, C. Michel, and M. Rueher, Refining Abstract Interpretation Based Value Analysis with Constraint Programming Techniques, Proceedings of the 18th international conference on Principles and practice of constraint programming (CP'12), pp.593-607
DOI : 10.1007/978-3-642-33558-7_43
URL : https://hal.archives-ouvertes.fr/hal-01099512

O. Ponsini, C. Michel, and M. Rueher, Verifying floating-point programs with constraint programming and abstract interpretation techniques, Automated Software Engineering, vol.34, issue.1, pp.191-217, 2016.
DOI : 10.1007/s10515-014-0154-2
URL : https://hal.archives-ouvertes.fr/hal-00860681

M. L. , P. , and J. Vignes, Etude statistique des erreurs dans l'arithmétique des ordinateurs ; application au contrôle des résultats d'algorithmes numériques, Numerische Mathematik, vol.23, pp.63-7275, 1974.

J. Puget and P. Van-hentenryck, A constraints satisfaction approach to a circuit design problem, Journal of Global Optimization, vol.13, issue.1, pp.75-93, 1998.
DOI : 10.1023/A:1008236911603

S. Ratschan, Approximate quantified constraint solving by cylindrical box decomposition Stefan Ratschan : Efficient solving of quantified inequality constraints over the real numbers, Reliable Computing ACM Transactions on Computational Logic, vol.894, issue.74, pp.21-42723, 2002.

D. Ratz, Box-Splitting-Strategien f??r den Intervall-Gauss-Seidel-Schritt in einem globalen Optimierungsverfahren, Computing, vol.1, issue.3-4, pp.337-353, 1994.
DOI : 10.1007/BF02307384

S. Hong, . Ryoo, and V. Nikolaos, Sahinidis : A branch-and-reduce approach to global optimization, Journal of Global Optimization, vol.8, issue.2, pp.107-138, 1996.

V. Nikolaos, M. Sahinidis, and . Tawarmalani, Baron 5.0 : Global optimisation of mixed-integer nonlinear programs. Rapport technique, 2002.

H. Schichl, M. C. Markót, and A. Neumaier, Exclusion regions for optimization problems, Journal of Global Optimization, vol.33, issue.4, pp.569-595, 2014.
DOI : 10.1007/s10898-013-0137-z

D. Hanif and H. Sherali-et-cihan, Tuncbilek : New reformulation linearization/convexification relaxations for univariate and multivariate polynomial programming problems, Operations Research Letters, vol.21, pp.1-9, 1997.

. Naum-zuselevich and . Shor, Dual quadratic estimates in polynomial and boolean programming, Annals of Operations Research, vol.25, pp.163-168, 1990.

M. Tawarmalani and V. Nikolaos, Sahinidis, éditeurs. Convexification and Global Optimization in Continuous and Mixed-Integer Non- Linear Programming, 2002.

C. Truchet, M. Pelleau, and F. Benhamou, Abstract Domains for Constraint Programming, with the Example of Octagons, 2010 12th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing, pp.72-79
DOI : 10.1109/SYNASC.2010.69
URL : https://hal.archives-ouvertes.fr/hal-01147921