Representations of positive polynomials on noncompact semialgebraic sets via KKT ideals

Journal of Pure and Applied Algebra

Volumn 209, Issue 1, 2007, Pages 189-200

  Demmel, James ^a   Nie, Jiawang ^a   Powers, Victoria ^b  

^a UNIVERSITY OF CALIFORNIA   (United States)

^b EMORY UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 33751547259     PISSN: 00224049     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.jpaa.2006.05.028     Document Type: Article

References (29)

1. Bochnak J., Coste M., and Roy M.-F. Real Algebraic Geometry (1998), Springer-Verlag, Berlin, Heidelberg
2. Cox D.A., Little J.B., and O'Shea D. Ideals, Varieties and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra. second edn. Undergraduate Texts in Mathematics (1997), Springer-Verlag, New York
3. Delzell C., and Prestel A. Positive Polynomials. Monographs in Mathematics (2001), Springer-Verlag, Berlin
4. Eisenbud D. Commutative Algebra with a View Toward Algebraic Geometry. Graduate Texts in Mathematics vol. 150 (1995), Springer-Verlag, New York
5. Eisenbud D. Computations in Algebraic Geometry with Macaulay 2. In: Grayson D.R., Stillman M., and Sturmfels B. (Eds). Algorithms and Computation in Mathematics (2002), Springer-Verlag, New York
6. Garey M.R., and Johnson D.S. Computers and Intractability: A Guide to the Theory of NP-Completeness (1979), W.H. Freeman and Company
7. Hanzon B., and Jibetean D. Global minimization of a multivariate polynomial using matrix methods. Journal of Global Optimization 27 (2003) 1-23
8. Henrion D., and Lasserre J. GloptiPoly: Global optimization over polynomials with Matlab and SeDuMi. ACM Transactions on Mathematical Software 29 (2003) 165-194
9. Henrion D., and Lasserre J. Detecting global optimality and extracting solutions in GloptiPoly. In: Henrion D., and Garulli A. (Eds). Positive Polynomials in Control. Lecture Notes on Control and Information Sciences (2005), Springer Verlag
10. Horst R., Pardalos P., and Thoai N. Introduction to Global Optimization. second edn (2000), Kluwer Academic Publishers
11. Jibetean D., and Laurent M. Semidefinite approximations for global unconstrained polynomial optimization. SIAM Journal on Optimization 16 2 (2005) 490-514
12. Lasserre J. Global optimization with polynomials and the problem of moments. SIAM Journal on Optimization 11 3 (2001) 796-817
13. M. Laurent, Semidefinite representations for finite varieties, Mathematical Programming (in press)
14. Nie J., and Demmel J. Minimum ellipsoid bounds for solutions of polynomial systems via sum of squares. Journal of Global Optimization 33 (2005) 511-525
15. Nie J., Demmel J., and Sturmfels B. Minimizing polynomials via sum of squares over the gradient ideal. Mathematical Programming 106 (2006) 587-606
16. J. Nie, M. Schweighofer, On the complexity of Putinar's Positivstellensatz, ArXiv:math.AG/0510309, 2005 (preprint)
17. Nocedal J., and Wright S.J. Numerical Optimization. Springer Series in Operations Research (1999), Springer-Verlag, New York
18. Parrilo P., and Sturmfels B. Minimizing polynomial functions. In: Basu S., and Gonzalez-Vega L. (Eds). Proceedings of the DIMACS Workshop on Algorithmic and Quantitative Aspects of Real Algebraic Geometry, Mathematics and Computer Science, March 2001 (2003), American Mathematical Society 83-100
19. Parrilo P. Semidefinite programming relaxations for semialgebraic problems. Mathematical Programming, Ser. B 96 2 (2003) 293-320
20. Powers V., and Reznick B. Polynomials positive on unbounded rectangles. Positive Polynomials in Control. Springer Lecture Notes in Control and Information Sciences vol. 312 (2005)
21. Prajna S., Papachristodoulou A., and Parrilo P. SOSTOOLS User's Guide. http://www.mit.edu/~parrilo/SOSTOOLS/
22. P. Parrilo, An explicit construction of distinguished representations of polynomials nonnegative over finite sets, IfA Technical Report AUT02-02, March 2002
23. Putinar M. Positive polynomials on compact semi-algebraic sets. Indiana University Mathematics Journal 42 (1993) 203-206
24. Schmüdgen K. The K-moment problem for compact semialgebraic sets. Mathematische Annalen 289 (1991) 203-206
25. Schweighofer M. Optimization of polynomials on compact semialgebraic sets. SIAM Journal on Optimization 15 3 (2005) 805-825
26. Shafarevich I.R. Basic Algebraic Geometry. die Grundlehren der mathematischen Wissenschaften, Band 213 (1974), Springer-Verlag, Berlin, Heidelberg
27. Stengle G. Complexity estimates for the Schmüdgen Positivstellensatz. Journal of Complexity 12 (1996) 167-174 MR 97d:14080
28. Sommese A.J., and Wampler C.W. The Numerical Solution of Systems of Polynomials (2005), World Scientific, Singapore
29. Vandenberghe L., and Boyd S. Semidefinite programming. SIAM Review 38 (1996) 49-95