A semidefinite relaxation scheme for multivariate quartic polynomial optimization with quadratic constraints

SIAM Journal on Optimization

Volumn 20, Issue 4, 2010, Pages 1716-1736

  Luo, Zhi Quan ^a   Zhang, Shuzhong ^b  

^a University of Minnesota   (United States)

^b CHINESE UNIVERSITY OF HONG KONG   (Hong Kong)

Author keywords

Approximation ratio; Quartic optimization; Randomized solution; Semidefinite programs relaxation

Indexed keywords

APPROXIMATE SOLUTION; APPROXIMATION RATIO; APPROXIMATION RATIOS; LINEAR CONSTRAINTS; NONCONVEX; OPTIMIZATION PROBLEMS; POLYNOMIAL TIME APPROXIMATION ALGORITHMS; POLYNOMIAL-TIME; POSITIVE DEFINITE; QUADRATIC CONSTRAINT; QUADRATIC OPTIMIZATION PROBLEMS; QUARTIC POLYNOMIAL; RANDOMIZED SOLUTIONS; RELAXATION SCHEMES; SEMI-DEFINITE MATRIX; SEMIDEFINITE PROGRAMS; SEMIDEFINITE RELAXATION;

APPROXIMATION ALGORITHMS; OPTIMIZATION;

POLYNOMIAL APPROXIMATION;

EID: 77951288110     PISSN: 10526234     EISSN: None     Source Type: Journal    
DOI: 10.1137/090772952     Document Type: Article

References (20)

1. W. Bär and F. Dittrich, Useful formula for moment computation of normal random variables with non-zero means, IEEE Trans. Automat. Control, 16 (1971), pp. 263-265.
2. A. Ben-Tal and A. Nemirovski, Robust convex optimization, Math. Oper. Res., 23 (1998), pp. 769-805.
3. A. B. Berkelaar, J. F. Sturm, and S. Zhang, Polynomial primal-dual cone affine scaling for semidefinite programming, Appl. Numer. Math., 29 (1999), pp. 317-333.
4. I. M. Bomze, M. Locatelli, and F. Tardella, New and old bounds for standard quadratic optimization: dominance, equivalence and incomparability, Math. Program., 115 (2008), pp. 31-64.
5. M. Goemans and D. Williamson, Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming, J. ACM, 42 (1995), pp. 1115-1145.
6. J. F. Cardoso, Blind signal separation: Statistical principles, Proc. IEEE, 90 (1998), pp. 2009-2026.
7. S. Kwapień and W. A. Woyczyński, Random Series and Stochastic Integrals: Single and Multiple, Birkhäuser, Boston, Cambridge, MA, 1992.
8. J. B. Lasserre, Global optimization with polynomials and the problem of moments, SIAM J. Optim., 11 (2001), pp. 796-817.
9. B. Maricic, Z.-Q. Luo, and T. N. Davidson, Blind constant modulus equalization via convex optimization, IEEE Trans. Signal Process., 51 (2003), pp. 805-818.
10. Z.-Q. Luo, N. D. Sidiropoulos, P. Tseng, and S. Zhang, Approximation bounds for quadratic optimization with homogeneous quadratic constraints, SIAM J. Optim., 18 (2007), pp. 1-28.
11. S. He, Z.-Q. Luo, J. W. Nie, and S. Zhang, Semidefinite relaxation bounds for indefinite homogeneous quadratic optimization, SIAM J. Optim., 19 (2008), pp. 503-523.
12. A. Nemirovski, C. Roos, and T. Terlaky, On maximization of quadratic form over intersection of ellipsoids with common center, Math. Program., 86 (1999), pp. 463-473.
13. Yu. Nesterov, Semidefinite relaxation and non-convex quadratic optimization, Optim. Methods Softw., 12 (1997), pp. 1-20.
14. Yu. Nesterov, Random walk in a simplex and quadratic optimization over convex polytopes, CORE discussion paper, The Catholic University of Leuven, Louvain-la-Neuve, Belgium, 2003.
15. P. A. Parrilo, Semidefinite programming relaxations for semialgebraic problems, Math. Program. Ser. B, 96 (2003), pp. 293-320.
16. L. Qi, Eigenvalues of a real supersymmetric tensor, J. Symbolic Comput., 40 (2005), pp. 1302-1324.
17. J. F. Sturm and S. Zhang, On cones of nonnegative quadratic functions, Math. Oper. Res., 28 (2003), pp. 246-267.
18. P. Biswas, T.-C. Liang, T.-C. Wang, and Y. Ye, Semidefinite programming based algorithms for sensor network localization, ACM Trans. Sensor Networks, 2 (2006), pp. 188-220.
19. A. M.-C. So, Y. Ye, and J. Zhang, A unified theorem on SDP rank reduction, Math. Oper.Res., 33 (2008), pp. 910-920.
20. V. Vazirani, Approximation algorithms, Springer, New York, 2001.