Erich Kaltofen
ABSTRACT
The problem of approximately factoring a real or complex multivariate polynomial f seeks minimal perturbations ? f to the coefficients of the input polynomial f so that the deformed polynomial f +Δ f has the desired factorization properties. Effcient algorithms exist that compute the nearest real or complex polynomial that has non-trivial factors (see [3,6 ]and the literature cited there). Here we consider the solution of the arising optimization problems polynomial optimization (POP)via semide finite programming (SDP). We restrict to real coe cients in the input and output polynomials.
- Borchers, B. CSDP,a C library for semide finite programming. Optimization Methods and Software 11 & 12 (1999), 613--623. Available at http://infohost.nmt.edu/~borchers/csdp.htmlGoogle Scholar
- Gao, S. Factoring multivariate polynomials via partial differential equations. Math. Comput. 72 242 (2003),801--822. Google ScholarDigital Library
- Gao, S., Kaltofen, E., May, J. P., Yang, Z., and Zhi, L. Approximate factorization of multivariate polynomials via differential equations. In ISSAC 2004 Proc. 2004 Internat. Symp. Symbolic Algebraic Comput. (New York,N.Y.,2004), J. Gutierrez, Ed., ACM Press,pp.167--174. Google ScholarDigital Library
- Hitz, M. A., Kaltofen, E., and Lakshman Y. N. E .cient algorithms for computing the nearest polynomial with a real root and related problems. In Proc. 1999 Internat. Symp. Symbolic Algebraic Comput. (ISSAC'99) (New York,N.Y., 1999), S. Dooley, Ed., ACM Press, pp.205--212. Google ScholarDigital Library
- Kaltofen, E., and May, J. On approximate irreducibility of polynomials in several variables. In ISSAC 2003 Proc. 2003 Internat. Symp. Symbolic Algebraic Comput. (New York,N.Y.,2003), J. R. Sendra, Ed., ACM Press, pp.161--168. Google ScholarDigital Library
- Kaltofen, E., May, J., Yang, Z., and Zhi, L. Approximate factorization of multivariate polynomials using singular value decomposition. Manuscript, 22 pages. Submitted, Jan.2006.Google Scholar
- Kaltofen, E., Yang, Z., and Zhi, L. Approximate greatest common divisors of several polynomials with linearly constrained coefficients and singular polynomials. In ISSAC MMVI Proc. 2006 Internat. Symp. Symbolic Algebraic Comput. (New York, N.Y., 2006), J.-G.Dumas, Ed., ACM Press, pp.169--176. Full version, 21 pages. Submitted, December 2006. Google ScholarDigital Library
- Lasserre, J. B. optimization with polynomials and the problem of moments. SIAM Journal on Optimization 11 (2001),796--817. Google ScholarDigital Library
- Nie, J., Demmel, J., and Gu, M. Global minimization of rational functions and the nearest GCDs. http://arxiv.org/pdf/math/0601110 2006.Google Scholar
- Prajna, S., Papachristodoulou, A., Seiler, P., and Parrilo, P. A. SOSTOOLS: Sum of squares optimization toolbox for MATLAB. Available from http://www.cds.caltech.edu/sostools and http://www.mit.edu/~parrilo/sostools 2004.Google Scholar
- Ruppert, W. M. Reduzibilität ebener Kurven.J. reine angew. Math. 369 (1986),167--191.Google Scholar
- Ruppert, W. M. Reducibility of polynomials f(x,y) modulo p J. Number Theory 77 (1999),62--70.Google ScholarCross Ref
- Waki, H., Kim, S., Kojima, M., and Muramatsu, M. SparsePOP: A sparse semide nite 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/SparsePOPGoogle Scholar
- Yamashita, M., Fujisawa, K., and Kojima, M. SDPARA: SemiDefinite Programming Algorithm PARAllel version. Parallel Computing 29 (2003),1053--1067. Available at http: min-//grid.r.dendai.ac.jp/sdpa/sdpa.6.00/sdpara.index.html Google ScholarDigital Library
Index Terms
- Lower bounds for approximate factorizations via semidefinite programming: (extended abstract)
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