Sparse SOS relaxations for minimizing functions that are summations of smal l polynomials

SIAM Journal on Optimization

Volumn 19, Issue 4, 2008, Pages 1534-1558

  Nie, Jiawang ^a   Demmel, James ^b  

^a UNIVERSITY OF CALIFORNIA   (United States)

^b UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

Nonlinear differential equations; Nonlinear least squares; Polynomial system; Polynomials; Sensor network localization; Sparsity; Sum of squares (SOS)

Indexed keywords

NON-LINEAR LEAST SQUARES; NONLINEAR DIFFERENTIAL EQUATION; POLYNOMIAL SYSTEM; POLYNOMIAL SYSTEMS; SENSOR NETWORK LOCALIZATION; SUM OF SQUARES;

COMMUNICATION CHANNELS (INFORMATION THEORY); COMPUTATIONAL COMPLEXITY; CONTROL THEORY; DIFFERENTIAL EQUATIONS; NUMERICAL METHODS; POLYNOMIALS; SENSOR NETWORKS;

NONLINEAR EQUATIONS;

EID: 70450210375     PISSN: 10526234     EISSN: None     Source Type: Journal    
DOI: 10.1137/060668791     Document Type: Article

References (29)

1. A. L. ALLGOWER, D. J. BATES, A. J. SOMMESE, AND C. W. WAMPLER, Solution of polynomial systems derived from differential equations, Computing, 76 (2005), pp, 1-10.
2. S. J. BENSON AND Y. YE, DSDP3: Dual scaling algorithm for general positive semidefinite programming, echnical report ANL/MCS-P851-1000, Mathematics and Computer Science Division, Argonne National Laboratory, 2001.
3. J. BLAIR AND B. PEYTON, An introduction to chordal graphs and clique trees, in Graph Theory and Sparse Matrix Computations, J. George, J. Gilbert, and J. Liu, eds., Springer Verlag, New York, 1993, pp. 1-30.
4. G. BLEKHERMAN, There are significantly more nonnegative polynomials than sums of squares, Israel J. Math., 153 (2006), pp. 355-380.
5. P. BISWAS, T. C. LIANG, K. C. TOH, T. C. WANG, AND Y. YE, Semidefinite programming approaches for sensor network localization with noisy distance measurements, IEEE Trans. Automat. Sci. Eng., 3 (2006), pp. 360-371.
6. A. R. CONN, N. I. M. GOULD, AND P. L. TOINT, Testing a class of methods for solving minimization problems with simple bounds on the variables, Math. Comp., 50 (1988), pp. 399-430.
7. R. E. CURTO AND L. A. FIALKOW, The truncated complex K-moment problem, Trans. Amer. Math. Soc., 352 (2000), pp. 2825-2855.
8. K. GATERMANN AND P. PARRILO, Symmetry groups, semidefinite programs, and sums of squares, J. Pure Appl. Algebra, 192 (2004), pp. 95-128.
9. D. HENRION AND J. LASSERRE, Detecting global optimality and extracting solutions in GloptiPoly, in Positive Polynomials in Control, D. Henrion and A. Garulli, eds., Lecture Notes on Control and Information Sciences, Springer Verlag, New York, 2005.
10. D. JIBETEAN AND M. LAURENT, Semidefinite approximations for global unconstrained polynomial optimization, SIAM J. Optim., 16 (2005), pp. 490-514.
11. S. KIM, M. KOJIMA, AND H. WAKI, Generalized Lagrangian duals and sums of squares relaxations of sparse polynomial optimization problems, SIAM J. Optim., 15 (2005), pp. 697-719.
12. M. KOJIMA, S. KIM, AND H. WAKI, Sparsity in sums of squares of polynomials, Math. Program., 103 (2005), pp. 45-62.
13. J. LASSERRE, Global optimization with polynomials and the problem of moments, SIAM J. Optim., 11 (2001), pp. 796-817.
14. J. LASSERRE, A sum of squares approximation of nonnegative polynomials, SIAM J. Optim., 16 (2006), pp. 751-765.
15. J. LASSERRE, Convergent SDP-relaxations in polynomial optimization with sparsity, SIAM J. Optim., 17 (2006), pp. 822-843. (Pubitemid 47265174)
16. M. LAURENT, Moment matrices and optimization over polynomials - A survey on selected topics, Emerging Applications of Algebraic Geometry, IMA Volumes in Mathematics and its Applications 149, (des. M. Putinar and S. Sullivant), Springer, to appear.
17. J. LÖFBERG, YALMIP: A toolbox for Modeling and Optimization in MATLAB, in Proceedings of the CACSD Conference, Taipei, Taiwan, 2004. http://control.ee.ethz.ch/~joloef/yalmip.php.
18. J. MORE, B. GARBOW, AND K. HILLSTROM, Testing unconstrained optimization software, ACM Trans. Math. Software, 7 (1981), pp. 17-41.
19. S. G. NASH, Newton-type minimization via the Lanczos method, SIAM J. Numer. Anal., 21 (1984), pp. 770-788.
20. J. NIE, J. DEMMEL, AND B. STURMFELS, Minimizing polynomials via sum of squares over the gradient ideal, Math. Program., Ser. A, 106 (2006), pp. 587-606.
21. J. NIE, Sum of squares methods for sensor network localization, Comput. Optim. Appl., to appear.
22. P. PARRILO AND B. STURMFELS, Minimizing polynomial functions, in Proceedings of the DI-MACS Workshop on Algorithmic and Quantitative Aspects of Real Algebraic Geometry in Mathematics and Computer Science 2001, S. Basu and L. Gonzalez-Vega, eds., American Mathematical Society, Providence, RI, 2003, pp. 83-100.
23. P. PARRILO, Semidefinite programming relaxations for semialgebraic problems, Math. Program. Ser. B, 96 (2003), pp. 293-320. (Pubitemid 44757842)
24. P. PARRILO, Exploiting structure in sum of squares programs, in Proceedings for the 42nd IEEE Conference on Decision and Control, Maui, Hawaii, 2003.
25. B. REZNICK, Extremal psd forms with few terms, Duke Math. J., 45 (1978), pp. 363-374.
26. M. SCHWEIGHOFER, Global optimization of polynomials using gradient tentacles and sums of squares, SIAM J. Optim., 17 (2006), pp. 920-942.
27. J. F. STURM, SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones, Optim. Methods Softw., 11 & 12 (1999), pp. 625-653.
28. H. WAKI, S. KIM, M. KOJIMA, AND M. MURAMATSU, Sums of squares and semidefinite programming relaxations for polynomial optimization problems with structured sparsity, SIAM J. Optim., 17 (2006), pp. 218-242.
29. H. WOLKOWICZ, R. SAIGAL, AND L. VANDENBERGHE, EDS., Handbook of semidefinite programming, Kluwer Publishers, Boston, 2000.