Discrete transforms, semidefinite programming, and sum-of-squares representations of nonnegative polynomials

SIAM Journal on Optimization

Volumn 16, Issue 4, 2006, Pages 939-964

  Roh, Tae ^a   Vandenberghe, Lieven ^a  

^a UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

Interior point methods; Nonnegative polynomials; Semidefinite programming

Indexed keywords

COSINE POLYNOMIALS; INTERIOR POINT METHODS; NONNEGATIVE POLYNOMIALS; SEMIDEFINITE PROGRAMMING;

ALGORITHMS; POLYNOMIAL APPROXIMATION; PROBLEM SOLVING;

MATHEMATICAL TRANSFORMATIONS;

EID: 33750925871     PISSN: 10526234     EISSN: None     Source Type: Journal    
DOI: 10.1137/040612646     Document Type: Article

References (34)

1. J. W. ADAMS, FIR digital filters with least-squares stopbands subject to peak-gain constraints, IEEE Trans. Circuits Syst., 39 (1991), pp. 376-388.
2. N. I. AHIEZER AND M. KREIN, Some Questions in the Theory of Moments, Transl. Math. Monogr. 2, AMS, Providence, RI, 1962 (translated from the 1938 Russian manuscript by W. Fleming and D. Prill).
3. F. ALIZADEH, J.-P. A. HAEBERLY, AND M. L. OVERTON, Primal-dual interior-point methods for semidefinite programming: Convergence rates, stability and numerical results, SIAM J. Optim., 8 (1998), pp. 746-768.
4. B. ALKIRE AND L. VANDENBERGHE, Convex optimization problems involving finite autocorrelation sequences, Math. Program. Ser. A, 93 (2002), pp. 331-359.
5. S. J. BENSON AND Y. YE, DSDP5: A Software Package Implementing the Dual-Scaling Algorithm for Semidefinite Programming, Technical report ANL/MCS-TM-255, Mathematics and Computer Science Division, Argonne National Laboratory, 2004.
6. S. J. BENSON, Y. YE, AND X. ZHANG, Solving large-scale sparse semidefinite programs for combinatorial optimization, SIAM J. Optim., 10 (2000), pp. 443-461.
7. W. L. BRIGGS AND V. E. HENSON, The DFT: An Owner's Manual for the Discrete Fourier Transform, SIAM, Philadelphia, PA, 1995.
8. T. N. DAVIDSON, Z.-Q. LUO, AND J. F. STURM, Linear matrix inequality formulation of spectral mask constraints, IEEE Trans. Signal Process., 50 (2002), pp. 2702-2715.
9. T. N. DAVIDSON, Z.-Q. LUO, AND K. M. WONG, Design of orthogonal pulse shapes for communications via semidefinite programming, IEEE Trans. Signal Process., 48 (2000), pp. 1433-1445.
10. P. J. DAVIS, Interpolation and Approximation, Dover, New York, 1975 (first published by Blaisdell Publishing Company in 1963).
11. B. DUMITRESCU, I. TABUS, AND P. STOICA, On the parametrization of positive real sequences and MA parameter estimation, IEEE Trans. Signal Process., 49 (2001), pp. 2630-2639.
12. L. FAYBUSOVICH, On Nesterov's approach to semi-infinite programming, Acta Appl. Math., 74 (2002), pp. 195-215.
13. L. FAYBUSOVICH, Semidefinite descriptions of cones defining spectral mask constraints, European J. Oper. Res., 169 (2006), pp. 1207-1221.
14. Y. GENIN, Y. HACHEZ, YU. NESTEROV, AND P. VAN DOOREN, Optimization problems over positive pseudopolynomial matrices, SIAM J. Matrix Anal. Appl., 25 (2003), pp. 57-79.
15. G. H. GOLUB AND J. H. WELSCH, Calculation of Gauss quadrature, rules, Math. Comp., 23 (1969), pp. 221-230.
16. Y. HACHEZ, Convex Optimization over Non-Negative Polynomials: Structured Algorithms and Applications, Ph.D. thesis, Université Catholique de Louvain, Belgium, 2003.
17. C. HELMBERG, F. RENDL, R. VANDERBEI, AND H. WOLKOWICZ, An interior-point method for semidefinite programming, SIAM J. Optim., 6 (1996), pp. 342-361.
18. D. HENRION, D. ARZELIER, AND D. PEAUCELLE, Positive polynomial matrices and improved LMI robustness conditions, Automatica, 39 (2003), pp. 1479-1485.
19. M. KOJIMA, S. SHINDOH, AND S. HARA, Interior-point methods for the monotone linear complementarity problem in symmetric matrices, SIAM J. Optim., 7 (1997), pp. 86-125.
20. M. G. KREIN AND A. A. NUDELMAN, The Markov Moment Problem and Extremal Problems, Transl. Math. Monogr. 50, AMS, Providence, RI, 1977.
21. J. B. LASSERRE, Global optimization with polynomials and the problem of moments, SIAM J. Optim., 11 (2001), pp. 796-817.
22. J. LÖFBERG AND P. A. PARRILO, From coefficients to samples: A new approach to SOS optimization, in Proceedings of the 43rd IEEE Conference on Decision and Control, 2004, pp. 3154-3159.
23. P. MOULIN, M. ANITESCU, K. O. KORTANEK, AND F. A. POTRA, The role of linear semi-infinite, programming in signal-adapted QMF bank design, IEEE Trans. Signal Process., 45 (1995), pp. 2160-2174.
24. Y. NESTEROV, Squared functional systems and optimization problems, in High Performance Optimization Techniques, J. Frenk, C. Roos, T. Terlaky, and S. Zhang, eds., Kluwer Academic, Norwell, MA, 2000, pp, 405-440.
25. P. A. PARRILO, Semidefinite programming relaxations for semialgebraic problems, Math. Program. Ser. B, 96 (2003), pp. 293-320.
26. A. STOICA, R. MOSES, AND P. STOICA, Enforcing positiveness on estimated spectral densities, Electron. Lett., 29 (1993), pp. 2009-2011.
27. P, STOICA, T. MCKELVEY, AND J. MARI, MA estimation in polynomial time, IEEE Trans. Signal Process., 48 (2000), pp. 1999-2012.
28. J. F. STURM, Using SEDUMI 1.02: A Matlab toolbox for optimization over symmetric canes, Optim. Methods Softw., 11-12 (1999), pp. 625-653.
29. G. SZEGÖ, Orthogonal Polynomials, AMS, Providence, RI, 2003 (first published in 1939).
30. M. J. TODD, K. C. TOH, AND R. H. TÜTÜNCÜ, On the Nesterov-Todd direction in semidefinite programming, SIAM J. Optim., 8 (1998), pp. 769-796.
31. R. H. TÜTÜNCÜ AND M. J. TODD, Solving semidefinitt:- quadratic-linear programs using SDPT3, Math. Program, Ser. B, 95 (2003), pp. 189-217.
32. P. P. VAIDYANATHAN, Multirate Systems and Filter Banks, Prentice-Hall, Englewood Cliffs, NJ, 1993.
33. L. VANDENBERGHE, V. R. BALAKRISHNAN, R. WALLIN, A. H. HANSOM, AND T. ROH, Interior point algorithms for semidefinite programming problems derived from the KYP lemrna, in Positive Polynomials in Control, D. Henrion and A. Garulli, eds., Lecture Notes in Control and Inform. Sci. 312, Springer, New York, 2005, pp. 195-238.
34. S.-P. WU, S. BOYD, AND L. VANDENBERGHE, FIR filter design via spectral factorization and convex optimization, in Applied and Computational Control, Signals, and Circuits. Vol. 1. B. Datta, ed., Birkhäuser, Boston, 1998, pp. 215-245.