Abstract
Errors in the coefficients due to floating point round-off or through physical measurement can render exact symbolic algorithms unusable. Hybrid symbolic-numeric algorithms compute minimal deformations of those coefficients that yield non-trivial results, e.g. polynomial factorizations or sparse interpolants. The question is: are the computed approximations the globally nearest to the input?
We present a new alternative to numerical optimization, namely the exact validation via symbolic methods of the global minimality of our deformations. Semidefinite programming and Newton refinement are used to compute a numerical sum-of-squares representation, which is converted to an exact rational identity for a nearby rational lower bound. Since the exact certificates leave no doubt, the numeric heuristics need not be fully analyzed. We demonstrate our approach on the approximate GCD, approximate factorization, and Rump’s model problems. The talk covers joint work with Bin Li, Zhengfeng Yang and Lihong Zhi.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Xavier Dahan and Éric Schost. Sharp estimates for triangular sets. In Jaime Gutierrez, editor, ISSAC 2004 Proc. 2004 Internat. Symp. Symbolic Algebraic Comput., pages 103–110, New York, N. Y., 2004. ACM Press.
Hazel Everett, Daniel Lazard, Sylvain Lazard, and Mohab Safey El Din. The Voronoi diagram of three lines in r 3. In SoCG ’07: Proceedings of the 23-rd Annual Symposium on Computational Geometry, pages 255–264. ACM, New York, USA, 2007.
G. H. Golub and C. F. Van Loan. Matrix Computations. Johns Hopkins University Press, Baltimore, Maryland, third edition, 1996.
Didier Henrion and Jean Bernard Lasserre. GloptiPoly: Global optimization over polynomials with Matlab and SeDuMi. ACM Trans. Math. Softw., 29(2):165–194, 2003.
David Jeffrey, editor. ISSAC 2008, New York, N. Y., 2008. ACM Press.
Erich Kaltofen, Bin Li, Kartik Sivaramakrishnan, Zhengfeng Yang, and Lihong Zhi. Lower bounds for approximate factorizations via semidefinite programming (extended abstract). In Verschelde and Watt [23], pages 203–204.
Erich Kaltofen, Bin Li, Zhengfeng Yang, and Lihong Zhi. Exact certification of global optimality of approximate factorizations via rationalizing sums-of-squares with floating point scalars. In Jeffrey [5], pages 155–163.
Erich Kaltofen, Bin Li, Zhengfeng Yang, and Lihong Zhi. Exact certification in global polynomial optimization via sums-of-squares of rational functions with rational coefficients, January 2009. Manuscript, 20 pages.
Erich Kaltofen, John May, Zhengfeng Yang, and Lihong Zhi. Approximate factorization of multivariate polynomials using singular value decomposition. J. Symbolic Comput., 43(5):359–376, 2008.
Erich Kaltofen, Zhengfeng Yang, and Lihong Zhi. Approximate greatest common divisors of several polynomials with linearly constrained coefficients and singular polynomials. In Jean-Guillaume Dumas, editor, ISSAC MMVI Proc. 2006 Internat. Symp. Symbolic Algebraic Comput., pages 169–176, New York, N. Y., 2006. ACM Press.
Jean B. Lasserre. Global SDP-relaxations in polynomial optimization with sparsity. SIAM J. on Optimization, 17(3):822–843, 2006.
Bin Li, Jiawang Nie, and Lihong Zhi. Approximate GCDs of polynomials and sparse SOS relaxations. Manuscript, 16 pages. Submitted, 2007.
J. Löfberg. YALMIP : A toolbox for modeling and optimization in MATLAB. In Proc. IEEE CCA/ISIC/CACSD Conf., Taipei, Taiwan, 2004. URL: http://control.ee.ethz.ch/joloef/yalmip.php.
M. Mignotte. Some useful bounds. In B. Buchberger, G. Collins, and R. Loos, editors, Computer Algebra, pages 259–263. Springer Verlag, Heidelberg, Germany, 2 edition, 1982.
Kosaku Nagasaka. Towards certified irreducibility testing of bivariate approximate polynomials. In T. Mora, editor, Proc. 2002 Internat. Symp. Symbolic Algebraic Comput. (ISSAC’02), pages 192–199, New York, N. Y., 2002. ACM Press.
Jiawang Nie and Markus Schweighofer. On the complexity of Putinar’s Positivstellensatz. J. Complexity, 23:135–70, 2007.
Helfried Peyrl and Pablo A. Parrilo. A Macaulay 2 package for computing sum of squares decompositions of polynomials with rational coefficients. In Verschelde and Watt [23], pages 207–208.
S. Prajna, A. Papachristodoulou, and P. A. Parrilo. SOSTOOLS: Sum of squares optimization toolbox for MATLAB. URL: http://www.cds.caltech.edu/sostools.
Siegfried M. Rump. Global optimization: a model problem, 2006. URL: http://www.ti3.tu-harburg.de/rump/Research/ModelProblem.pdf.
Siegfried M. Rump and H. Sekigawa. The ratio between the Toeplitz and the unstructured condition number, 2006. To appear. URL: http://www.ti3.tu-harburg.de/paper/rump/RuSe06.pdf.
Mohab Safey El Din. Computing the global optimum of a multivariate polynomial over the reals. In Jeffrey [5].
Jos F. Sturm. Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Optimization Methods and Software, 11/12:625–653, 1999.
Jan Verschelde and Stephen M. Watt, editors. SNC’07 Proc. 2007 Internat. Workshop on Symbolic-Numeric Comput., New York, N. Y., 2007. ACM Press.
Hayato Waki, Sunyoung Kim, Masakazu Kojima, and Masakazu Muramatsu. Sparse-POP: A sparse semidefinite programming relaxation of polynomial optimization problems. Research Report B-414, Tokyo Institute of Technology, Dept. of Mathematical and Computing Sciences, Oh-Okayama, Meguro 152-8552, Tokyo, Japan, 2005. Available at http://www.is.titech.ac.jp/kojima/SparsePOP.
Henry Wolkowicz, Romesh Saigal, and Lieven (Eds.) Vandenberghe. Handbook of Semidefinite Programming. Kluwer Academic, Boston, 2000.
Ting Zhang, Rong Xiao, and Bican Xia. Real soultion isolation based on interval Krawczyk operator. In Sung il Pae and Hyungju Park, editors, Proc. of the Seventh Asian Symposium on Computer Mathematics, pages 235–237, Seoul, South Korea, 2005. Korea Institute for Advanced Study. Extended abstract.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Vienna
About this chapter
Cite this chapter
Kaltofen, E. (2009). Exact Certification in Global Polynomial Optimization Via Rationalizing Sums-Of-Squares. In: Robbiano, L., Abbott, J. (eds) Approximate Commutative Algebra. Texts and Monographs in Symbolic Computation. Springer, Vienna. https://doi.org/10.1007/978-3-211-99314-9_9
Download citation
DOI: https://doi.org/10.1007/978-3-211-99314-9_9
Published:
Publisher Name: Springer, Vienna
Print ISBN: 978-3-211-99313-2
Online ISBN: 978-3-211-99314-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)