article

Formalization of Bernstein Polynomials and Applications to Global Optimization

Authors:

Anthony NarkawiczAuthors Info & Claims

Journal of Automated Reasoning, Volume 51, Issue 2

Pages 151 - 196

https://doi.org/10.1007/s10817-012-9256-3

Published: 01 August 2013 Publication History

Abstract

This paper presents a formalization in higher-order logic of a practical representation of multivariate Bernstein polynomials. Using this representation, an algorithm for finding lower and upper bounds of the minimum and maximum values of a polynomial has been formalized and verified correct in the Prototype Verification System (PVS). The algorithm is used in the definition of proof strategies for formally and automatically solving polynomial global optimization problems.

References

[1]

Akbarpour, B., Paulson, L.C.: MetiTarski: an automatic theorem prover for real-valued special functions. J. Autom. Reason. 44(3), 175-205 (2010)

Digital Library

[2]

Alford, J.: Translation of Bernstein coefficients under an affine mapping of the unit interval. Technical Memorandum NASA/TM-2012-217557, NASA Langley Research Center (2012)

[3]

Archer, M., Di Vito, B., Muñoz, C. (eds.): Design and Application of Strategies/Tactics in Higher Order Logics. No. NASA/CP-2003-212448, NASA, Langley Research Center, Hampton VA 23681-2199, USA (2003)

[4]

Bertot, Y., Guilhot, F., Mahboubi, A.: A formal study of Bernstein coefficients and polynomials. Tech. Rep. INRIA-005030117, INRIA (2010)

[5]

Brisebarre, N., Joldes, M., Martin-Dorel, É., Mayero, M., Muller, J.M., Pasca, I., Rideau, L., Théry, L.: Rigorous polynomial approximation using Taylor models in Coq. In: Goodloe, A., Person, S. (eds.) Proceedings of the NASA Formal Methods Symposium (NFM 2012). Lecture Notes in Computer Science, vol. 7226, pp. 85-99. Springer, Norfolk, US (2012)

[6]

de Casteljau, P.: Formes à pôles. Hermès (1985)

[7]

Cháves, F., Daumas, M.: A library of Taylor models for PVS automatic proof checker. Technical Report RR2006-07, École Normale Supérieure de Lyon (2006)

[8]

Cohen, C., Mahboubi, A.: Formal proofs in real algebraic geometry: from ordered fields to quantifier elimination. Logical Methods in Computer Science (LMCS) 8(1:02), 1-40 (2012)

[9]

Collins, G.: Quantifier elimination for real closed fields by cylindrical algebraic decomposition. In: Second GI Conference on Automata Theory and Formal Languages. Lecture Notes in Computer Science, vol. 33, pp. 134-183. Springer, Kaiserslautern (1975)

Digital Library

[10]

Crespo, L.G., Muñoz, C.A., Narkawicz, A.J., Kenny, S.P., Giesy, D.P.: Uncertainty analysis via failure domain characterization: polynomial requirement functions. In: Proceedings of European Safety and Reliability Conference. Troyes, France (2011)

[11]

Daumas, M., Lester, D., Muñoz, C.: Verified real number calculations: a library for interval arithmetic. IEEE Trans. Comput. 58(2), 1-12 (2009)

[12]

Delahaye, D., Mayero, M.: Field, une procédure de décision pour les nombres réels en coq. In: Castéran, P. (ed.) Journées Francophones des Langages Applicatifs (JFLA'01), pp. 33-48. Collection Didactique, INRIA, Pontarlier, France, Janvier (2001)

[13]

Di Vito, B.: Manip user's guide, version 1.3. Technical Memorandum NASA/TM-2002-211647, NASA Langley Research Center (2002)

[14]

de Dinechin, F., Lauter, C., Melquiond, G.: Certifying the floating-point implementation of an elementary function using Gappa. IEEE Trans. Comput. 60(2), 242-253 (2011)

Digital Library

[15]

Garloff, J.: Convergent bounds for the range of multivariate polynomials. In: Proceedings of the International Symposium on interval mathematics on Interval mathematics 1985, pp. 37-56. Springer-Verlag, London, UK (1985)

[16]

Garloff, J.: The Bernstein algorithm. Interval Comput. 4, 154-168 (1993)

[17]

Garloff, J.: Application of Bernstein expansion to the solution of control problems. Reliab. Comput. 6, 303-320 (2000)

[18]

Granvilliers, L., Benhamou, F.: RealPaver: an interval solver using constraint satisfaction techniques. ACM Trans. Math. Softw. 32(1), 138-156 (2006)

Digital Library

[19]

Harrison, J.: Metatheory and reflection in theorem proving: a survey and critique. Technical Report CRC-053, SRI Cambridge, Millers Yard, Cambridge, UK (1995)

[20]

Harrison, J.: Verifying nonlinear real formulas via sums of squares. In: Theorem Proving in Higher Order Logics. Lecture Notes in Computer Science, vol. 4732, pp. 102-118. Springer (2007)

[21]

Hunt, W.A., Jr., Krug, R.B., Moore, J.S.: Linear and nonlinear arithmetic in ACL2. In: Geist, D., Tronci, E. (eds.) Proceedings of Correct Hardware Design and Verification Methods (CHARME). Lecture Notes in Computer Science, vol. 2860, pp. 319-333. Springer, L'Aquila, Italy (2003)

[22]

Kaltofen, E.L., Li, B., Yang, Z., Zhi, L.: Exact certification in global polynomial optimization via sums-of-squares of rational functions with rational coefficients. In: Robbiano, L., Abbott, J. (eds.) Approximate Commutative Algebra. Texts and Monographs in Symbolic Computation, Springer Vienna (2010)

[23]

Kuchar, J., Yang, L.: A review of conflict detection and resolution modeling methods. IEEE Trans. Intell. Transp. Syst. 1(4), 179-189 (2000)

Digital Library

[24]

Lorentz, G.G.: Bernstein Polynomials, 2nd edn. Chelsea Publishing Company, New York, N.Y. (1986)

[25]

Mahboubi, A.: Implementing the cylindrical algebraic decomposition within the Coq system. Math. Struct. Comput. Sci. 17(1), 99-127 (2007)

Digital Library

[26]

McLaughlin, S., Harrison, J.: A proof-producing decision procedure for real arithmetic. In: Nieuwenhuis, R. (ed.) Proceedings of the 20th International Conference on Automated Deduction, proceedings. Lecture Notes in Computer Science, vol. 3632, pp. 295-314 (2005)

[27]

Melquiond, G.: Proving bounds on real-valued functions with computations. In: Armando, A., Baumgartner, P., Dowek, G. (eds.) Automated Reasoning, 4th International Joint Conference, IJCAR 2008, Sydney, Australia, August 12-15, 2008, Proceedings. Lecture Notes in Computer Science, vol. 5195, pp. 2-17. Springer (2008).

[28]

Moa, B.: Interval methods for global optimization. Ph.D. thesis, University of Victoria (2007)

[29]

Monniaux, D., Corbineau, P.: On the generation of Positivstellensatz witnesses in degenerate cases. In: Proceedings of Interactive Theorem Proving (ITP). Lecture Notes in Computer Science (2011)

[30]

Muñoz, C., Mayero, M.: Real automation in the field. Tech. Rep. NASA/CR-2001-211271 Interim ICASE Report No. 39, ICASE-NASA Langley, ICASE Mail Stop 132C, NASA Langley Research Center, Hampton VA 23681-2199, USA (2001)

[31]

Nataraj, P.S.V., Arounassalame, M.: A new subdivision algorithm for the Bernstein polynomial approach to global optimization. International Journal of Automation and Computing (IJAC) 4(4), 342-352 (2007)

[32]

Neumaier, A.: Taylor forms - use and limits. Reliab. Comput. 9(1), 43-79 (2003)

[33]

Owre, S., Rushby, J., Shankar, N.: PVS: a prototype verification system. In: Kapur, D. (ed.) Proceeding of the 11th International Conference on Automated Deductioncade. Lecture Notes in Artificial Intelligence, vol. 607, pp. 748-752. Springer (1992)

[34]

Parrilo, P.A.: Semidefinite programming relaxations for semialgebraic problems.Math. Program. 96, 293-320 (2003).

[35]

Passmore, G.O.: Combined decision procedures for nonlinear arithmetics, real and complex. Ph.D. thesis, The Univesity of Edinburgh (2011)

[36]

Passmore, G.O., Jackson, P.B.: Combined decision techniques for the existential theory of the reals. In: Dixon, L. (ed.) Proceedings of Calculemus/Mathematical Knowledge Managment. LNAI, pp. 122-137, no. 5625. Springer-Verlag (2009)

[37]

Ray, S., Nataraj, P.S.: An efficient algorithm for range computation of polynomials using the Bernstein form. J. Glob. Optim. 45, 403-426 (2009)

Digital Library

[38]

Smith, A.P.: Fast construction of constant bound functions for sparse polynomials. J. Glob. Optim. 43, 445-458 (2009)

Digital Library

[39]

Tarski, A.: A Decision Method for Elementary Algebra and Geometry. Univeristy of California Press (1951)

[40]

Verschelde, J.: The PHC pack, the database of polynomial systems. Tech. rep., Univeristy of Illinois, Mathematics Department, Chicago, IL (2001)

[41]

Zippel, R.: Effective Polynomial Computation. Kluwer Academic Publishers (1993)

[42]

Zumkeller, R.: Formal global optimisation with Taylor models. In: Furbach, U., Shankar, N. (eds.) Proceedings of the Third International Joint Conference on Automated Reasoning. Lecture Notes in Computer Science, vol. 4130, pp. 408-422 (2006)

[43]

Zumkeller, R.: Global Optimization in Type Theory. Ph.D. thesis, École Polytechnique Paris (2008)

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Formalization of Bernstein Polynomials and Applications to Global Optimization
1. Theory of computation
  1. Design and analysis of algorithms

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cover image Journal of Automated Reasoning

Journal of Automated Reasoning Volume 51, Issue 2

August 2013

110 pages

ISSN:0168-7433

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Copyright © Copyright © 2013 Springer Science+Business Media Dordrecht.

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Springer-Verlag

Berlin, Heidelberg

Publication History

Published: 01 August 2013

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