Validated Constraint Solving—Practicalities, Pitfalls, and New Developments
- R. Baker Kearfott
- … show all 1 hide
Rent the article at a discountRent now
* Final gross prices may vary according to local VAT.Get Access
Many constraint propagation techniques iterate through the constraints in a straightforward manner, but can fail because they do not take account of the coupling between the constraints.However, some methods of taking account of this coupling are local in nature, and fail if the initial search region is too large.We put into perspective newer methods, based on linear relaxations, that can often replace brute-force search with the solution of a large, sparse linear program.
Robustness has been recognized as important in geometric computations and elsewhere for at least a decade, and more and more developers are including validation in the design of their systems. We provide citations to our work and to the work of others to-date in developing validated versions of linear relaxations.
This work is in the form of a brief review and prospectus for future development. We give various simple examples to illustrate our points.
- Babichev, A. B., Kadyrova, O. B., Kashevarova, T. P., Leshchenko, A. S., and Semenov, A. L.: UniCalc, A Novel Approach to Solving Systems of Algebraic Equations, Interval Computations 2 (1993), pp. 29–47.
- Benhamou, F. Interval Constraints, Interval Propagation. In: Floudas, C., Pardalos, P. eds. (2001) Encyclopedia of Optimization. Kluwer Academic Publishers, Dordrecht
- Berz, M.: Cosy Infinity Web Page, 2000, http://cosy.pa.msu.edu/.
- Cleary, J. G. (1987) Logical Arithmetic. Future Computing Systems 2: pp. 125-149
- Floudas, C. A. (2000) Deterministic Global Optimization: Theory, Algorithms and Applications. Kluwer Academic Publishers, Dordrecht
- Hongthong, S. and Kearfott, R. B.: Rigorous Linear Overestimators and Underestimators, preprint, 2004, http://interval.louisiana.edu/preprints/estimates_of_powers.pdf.
- Jansson, Ch. (2004) A Rigorous Lower Bound for the Optimal Value of Convex Optimization Problems. J. Global Optim. 28: pp. 121-137 CrossRef
- Kearfott, R. B. (1991) Decomposition of Arithmetic Expressions to Improve the Behavior of Interval Iteration for Nonlinear Systems. Computing 47: pp. 169-191
- Kearfott, R. B.: Empirical Comparisons of Linear Relaxations and Alternate Techniques in Validated Deterministic Global Optimization, preprint, 2004, http://interval.louisiana.edu/preprints/validated_global_optimization_search_comparisons.pdf.
- Kearfott, R. B.: Globsol: History, Composition, and Advice on Use, in: Global Optimization and Constraint Satisfaction, Lecture Notes in Computer Science, Springer-Verlag, New York, 2003, pages 17–31.
- Kearfott, R. B.: Interval Analysis: Interval Newton Methods, in: Encyclopedia of Optimization, volume 3, Kluwer Academic Publishers, 2001, pp. 76–78.
- Kearfott, R. B. (1996) Rigorous Global Search: Continuous Problems. Kluwer Academic Publishers, Dordrecht
- Kearfott, R. B. and Hongthong, S.: A Preprocessing Heuristic for Determining the Difficulty of and Selecting a Solution Strategy for Nonconvex Optimization, preprint, 2003, http://interval.louisiana.edu/preprints/2003_symbolic_analysis_of_GO.pdf.
- Kearfott, R. B., Neher, M., Oishi, S., and Rico, F.: Libraries, Tools, and Interactive Systems for Verified Computations: Four Case Studies, in: Alt, R., Frommer, A., Kearfott, R. B., and Luther, W. (eds), Numerical Software with Result Verification, Lecture Notes in Computer Science 2991, Springer-Verlag, New York, 2004, pp. 36–63.
- Kearfott, R. B., Walster, G.W. (2002) Symbolic Preconditioning with Taylor Models: Some Examples. Reliable Computing 8: pp. 453-468 CrossRef
- Kreinovich, V., Lakeyev, A., Rohn, J., Kahl, P. (1998) Computational Complexity and Feasibility of Data Processing and Interval Computations. Kluwer Academic Publishers, Dordrecht
- Lebbah, Y., Michel, C., Rueher, M., Daney, D., and Merlet, J.-P.: Efficient and Safe Global Constraints for Handling Numerical Constraint Systems, SIAM J. Numer. Anal., accepted for publication.
- Narin’yani, A. S. (1991) Intelligent Software Technology for the New Decade. Comm. ACM 34: pp. 60-67 CrossRef
- Neumaier, A. (1990) Interval Methods for Systems of Equations. Cambridge University Press, Cambridge
- Neumaier, A. and Shcherbina, O.: Safe Bounds in Linear and Mixed-Integer Programming, Math. Prog. 99 (2) (2004), pp. 283–296, http://www.mat.univie.ac.at/~neum/ms/mip.pdf.
- Rump, S. M. et al.: INTLAB home page, 2000, http://www.ti3.tu-harburg.de/~rump/intlab/index.html.
- Tawarmalani, M., Sahinidis, N. V. (2002) Convexification and Global Optimization in Continuous and Mixed-Integer Nonlinear Programming: Theory, Algorithms, and Applications. Kluwer Academic Publishers, Dordrecht
- Van Hentenryck, P., Michel, L., Deville, Y. (1997) Numerica: A Modeling Language for Global Optimization. MIT Press, Cambridge
- Validated Constraint Solving—Practicalities, Pitfalls, and New Developments
Volume 11, Issue 5 , pp 383-391
- Cover Date
- Print ISSN
- Online ISSN
- Kluwer Academic Publishers-Plenum Publishers
- Additional Links
- Author Affiliations
- 1. University of Louisiana, Lafayette, Louisiana, 70504-1010, USA