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Certification of Bounds of Non-linear Functions: The Templates Method

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Intelligent Computer Mathematics (CICM 2013)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 7961))

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Abstract

The aim of this work is to certify lower bounds for real-valued multivariate functions, defined by semialgebraic or transcendental expressions. The certificate must be, eventually, formally provable in a proof system such as Coq. The application range for such a tool is widespread; for instance Hales’ proof of Kepler’s conjecture yields thousands of inequalities. We introduce an approximation algorithm, which combines ideas of the max-plus basis method (in optimal control) and of the linear templates method developed by Manna et al. (in static analysis). This algorithm consists in bounding some of the constituents of the function by suprema of quadratic forms with a well chosen curvature. This leads to semialgebraic optimization problems, solved by sum-of-squares relaxations. Templates limit the blow up of these relaxations at the price of coarsening the approximation. We illustrate the efficiency of our framework with various examples from the literature and discuss the interfacing with Coq.

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References

  1. Hales, T.C., Harrison, J., McLaughlin, S., Nipkow, T., Obua, S., Zumkeller, R.: A revision of the proof of the kepler conjecture. Discrete & Computational Geometry 44(1), 1–34 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  2. Hales, T.C.: Introduction to the flyspeck project. In: Coquand, T., Lombardi, H., Roy, M.-F. (eds.) Mathematics, Algorithms, Proofs. Dagstuhl Seminar Proceedings, vol. 05021, Internationales Begegnungs- und Forschungszentrum für Informatik (IBFI), Schloss Dagstuhl, Germany (2006)

    Google Scholar 

  3. Peyrl, H., Parrilo, P.A.: Computing sum of squares decompositions with rational coefficients. Theor. Comput. Sci. 409(2), 269–281 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. 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. JSC 47(1), 1–15 (2012); In memory of Wenda Wu (1929–2009)

    Google Scholar 

  5. Monniaux, D., Corbineau, P.: On the generation of Positivstellensatz witnesses in degenerate cases. In: van Eekelen, M., Geuvers, H., Schmaltz, J., Wiedijk, F. (eds.) ITP 2011. LNCS, vol. 6898, pp. 249–264. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  6. Besson, F.: Fast reflexive arithmetic tactics the linear case and beyond. In: Altenkirch, T., McBride, C. (eds.) TYPES 2006. LNCS, vol. 4502, pp. 48–62. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  7. Harrison, J.: Verifying nonlinear real formulas via sums of squares. In: Schneider, K., Brandt, J. (eds.) TPHOLs 2007. LNCS, vol. 4732, pp. 102–118. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  9. Solovyev, A., Hales, T.C.: Formal verification of nonlinear inequalities with taylor interval approximations. CoRR, abs/1301.1702 (2013)

    Google Scholar 

  10. Gao, S., Avigad, J., Clarke, E.M.: Delta-complete decision procedures for satisfiability over the reals. CoRR, abs/1204.3513 (2012)

    Google Scholar 

  11. Lasserre, J.B.: Global optimization with polynomials and the problem of moments. SIAM Journal on Optimization 11(3), 796–817 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  12. Parrilo, P.A., Sturmfels, B.: Minimizing polynomial functions. DIMACS Ser. Discrete Math. Theoret. Comput. Sci, vol. 60, pp. 83–99. Amer. Math. Soc., Providence (2003)

    Google Scholar 

  13. Waki, H., Kim, S., Kojima, M., Muramatsu, M.: Sums of squares and semidefinite programming relaxations for polynomial optimization problems with structured sparsity. SIAM Journal on Optimization 17, 218–242 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  14. Zumkeller, R.: Rigorous Global Optimization. PhD thesis, Ècole Polytechnique (2008)

    Google Scholar 

  15. Hansen, E.R.: Sharpening interval computations. Reliable Computing 12(1), 21–34 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  16. Cartis, C., Gould, N.I.M., Toint, P.L.: Adaptive cubic regularisation methods for unconstrained optimization. part i: motivation, convergence and numerical results. Math. Program. 127(2), 245–295 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. Montaz Ali, M., Khompatraporn, C., Zabinsky, Z.B.: A numerical evaluation of several stochastic algorithms on selected continuous global optimization test problems. J. of Global Optimization 31(4), 635–672 (2005)

    Article  MATH  Google Scholar 

  18. Sankaranarayanan, S., Sipma, H.B., Manna, Z.: Scalable analysis of linear systems using mathematical programming. In: Cousot, R. (ed.) VMCAI 2005. LNCS, vol. 3385, pp. 25–41. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  19. Adje, A., Gaubert, S., Goubault, E.: Coupling policy iteration with semi-definite relaxation to compute accurate numerical invariants in static analysis. Logical Methods in Computer Science 8(1), 1–32 (2012)

    Article  MathSciNet  Google Scholar 

  20. Fleming, W.H., McEneaney, W.M.: A max-plus-based algorithm for a Hamilton-Jacobi-Bellman equation of nonlinear filtering. SIAM J. Control Optim. 38(3), 683–710 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  21. Akian, M., Gaubert, S., Lakhoua, A.: The max-plus finite element method for solving deterministic optimal control problems: basic properties and convergence analysis. SIAM J. Control Optim. 47(2), 817–848 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  22. McEneaney, W.M., Deshpande, A., Gaubert, S.: Curse-of-complexity attenuation in the curse-of-dimensionality-free method for HJB PDEs. In: Proc. of the 2008 American Control Conference, Seattle, Washington, USA, pp. 4684–4690 (June 2008)

    Google Scholar 

  23. McEneaney, W.M.: A curse-of-dimensionality-free numerical method for solution of certain HJB PDEs. SIAM J. Control Optim. 46(4), 1239–1276 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  24. Gaubert, S., McEneaney, W.M., Qu, Z.: Curse of dimensionality reduction in max-plus based approximation methods: Theoretical estimates and improved pruning algorithms. In: CDC-ECC [36], pp. 1054–1061

    Google Scholar 

  25. Allamigeon, X., Gaubert, S., Magron, V., Werner, B.: Certification of inequalities involving transcendental functions: combining sdp and max-plus approximation. To appear in the Proceedings of the European Control Conference, ECC 2013, Zurich (2013)

    Google Scholar 

  26. Lasserre, J.B., Putinar, M.: Positivity and optimization for semi-algebraic functions. SIAM Journal on Optimization 20(6), 3364–3383 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  27. Yamashita, M., Fujisawa, K., Nakata, K., Nakata, M., Fukuda, M., Kobayashi, K., Goto, K.: A high-performance software package for semidefinite programs: Sdpa7. Technical report, Dept. of Information Sciences, Tokyo Institute of Technology, Tokyo, Japan (2010)

    Google Scholar 

  28. Akian, M., Gaubert, S., Kolokoltsov, V.N.: Set coverings and invertibility of functional galois connections. In: Litvinov, G.L., Maslov, V.P. (eds.) Idempotent Mathematics and Mathematical Physics, Contemporary Mathematics, pp. 19–51. American Mathematical Society (2005)

    Google Scholar 

  29. Calafiore, G., Dabbene, F.: Reduced vertex set result for interval semidefinite optimization problems. Journal of Optimization Theory and Applications 139, 17–33 (2008), doi:10.1007/s10957-008-9423-1

    Article  MathSciNet  MATH  Google Scholar 

  30. Riener, C., Theobald, T., Andrén, L.J., Lasserre, J.B.: Exploiting symmetries in sdp-relaxations for polynomial optimization. CoRR, abs/1103.0486 (2011)

    Google Scholar 

  31. Waki, H., Kim, S., Kojima, M., Muramatsu, M., Sugimoto, H.: Algorithm 883: Sparsepop—a sparse semidefinite programming relaxation of polynomial optimization problems. ACM Trans. Math. Softw. 35(2) (2008)

    Google Scholar 

  32. Montanher, T.M.: Intsolver: An interval based toolbox for global optimization, Version 1.0 (2009), www.mathworks.com

  33. Hansen, E.R., Greenberg, R.I.: An interval newton method. Applied Mathematics and Computation 12(2-3), 89–98 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  34. Brisebarre, N., Joldeş, M.: Chebyshev interpolation polynomial-based tools for rigorous computing. In: Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation, ISSAC 2010, pp. 147–154. ACM, New York (2010)

    Chapter  Google Scholar 

  35. Lasserre, J.B., Thanh, T.P.: Convex underestimators of polynomials. In: CDC-ECE [36], pp. 7194–7199

    Google Scholar 

  36. Proceedings of the 50th IEEE Conference on Decision and Control and European Control Conference, CDC-ECC 2011, Orlando, FL, USA, December 12-15. IEEE (2010)

    Google Scholar 

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Author information

Authors and Affiliations

  1. INRIA and CMAP, École Polytechnique, Palaiseau, France

    Xavier Allamigeon & Stéphane Gaubert

  2. INRIA and LIX, École Polytechnique, Palaiseau, France

    Victor Magron & Benjamin Werner

Authors
  1. Xavier Allamigeon
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  2. Stéphane Gaubert
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  3. Victor Magron
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  4. Benjamin Werner
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Editor information

Editors and Affiliations

  1. Computing and Software, McMaster University, 1280 Main Street West, L8S 4K1, Hamilton, ON, Canada

    Jacques Carette

  2. School of Informatics, University of Edinburgh, 10 Crichton Street, EH8 9AB, Edinburgh, Scotland, UK

    David Aspinall

  3. School of Computer Science, University of Birmingham, Edgbaston, B15 2TT, Birmingham, UK

    Christoph Lange

  4. Faculty of Informatics, Department of Computer Graphics and Design, Masaryk University, Botanická 68a, 60200, Brno, Czech Republic

    Petr Sojka

  5. Johannes Kepler University Linz (JKU), RISC, Altenberger Strasse 69, 4040, Linz, Austria

    Wolfgang Windsteiger

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Allamigeon, X., Gaubert, S., Magron, V., Werner, B. (2013). Certification of Bounds of Non-linear Functions: The Templates Method. In: Carette, J., Aspinall, D., Lange, C., Sojka, P., Windsteiger, W. (eds) Intelligent Computer Mathematics. CICM 2013. Lecture Notes in Computer Science(), vol 7961. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39320-4_4

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  • DOI: https://doi.org/10.1007/978-3-642-39320-4_4

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