Biquadratic optimization over unit spheres and semidefinite programming relaxations

SIAM Journal on Optimization

Volumn 20, Issue 3, 2009, Pages 1286-1310

  Ling, Chen ^a,b   Nie, Jiawang ^c   Qi, Liqun ^b   Ye, Yinyu ^b,d  

^a ZHEJIANG UNIVERSITY OF FINANCE AND ECONOMICS   (China)

^b HONG KONG POLYTECHNIC UNIVERSITY   (Hong Kong)

^c UNIVERSITY OF CALIFORNIA   (United States)

^d Stanford University   (United States)

Author keywords

Approximate solution; Biquadratic optimization; Polynomial time approximation scheme; Semidefinite programming; Sum of squares

Indexed keywords

APPROXIMATE SOLUTION; APPROXIMATION BOUNDS; APPROXIMATION METHODS; EIGENVALUE METHODS; ERROR BOUND; FIRST ORDER; NP-HARD; NUMERICAL EXAMPLE; POLYNOMIAL TIME APPROXIMATION SCHEMES; POLYNOMIAL-TIME ALGORITHMS; SDP RELAXATION; SEMI-DEFINITE PROGRAMMING; SEMIDEFINITE PROGRAMMING RELAXATIONS; SUM OF SQUARES; THEORETICAL RESULT;

APPROXIMATION ALGORITHMS; COMPUTATIONAL COMPLEXITY; EIGENVALUES AND EIGENFUNCTIONS; MATHEMATICAL PROGRAMMING;

POLYNOMIAL APPROXIMATION;

EID: 73249147288     PISSN: 10526234     EISSN: None     Source Type: Journal    
DOI: 10.1137/080729104     Document Type: Article

References (40)

1. I. M. BOMZE AND E. DE KLERK, Solving standard quadratic optimization problems via linear, semidefinite and copositive programming, J. Global Optim., 24 (2002), pp. 163-185.
2. A. P. CALDERÓN, A note on bi-quadratic forms, Linear Algebra Appl., 7 (1973), pp. 175-177.
3. J. F. CARDOSO, High-order contrasts for independent component analysis, Neural Comput., 11 (1999), pp. 157-192.
4. M.-D. CHOI, Positive semidefinite biquadratic forms, Linear Algebra Appl., 12 (1975), pp. 95-100.
5. P. COMON, Independent component analysis, a new concept?, Signal Process., 36 (1994), pp. 287-314.
6. G. DAHL, J. M. LEINAAS, J. MYRHEIM, AND E. OVRUM, A tensor product matrix approximation problem in quantum physics, Linear Algebra Appl., 420 (2007), pp. 711-725.
7. L. DE LATHAUWER, P. COMON, B. DE MOOR, AND J. VANDEWALLE, Higher-order power method-application in independent component analysis, in Proceedings of the International Symposium on Nonlinear Theory and its Applications (NOLTA'95), Las Vegas, NV, 1995, IEICE, pp. 91-96.
8. N) approximation of higher-order tensors, SIAM J. Matrix Anal. Appl., 21 (2000), pp. 1324-1342.
9. J. DEMMEL, J. NIE, AND V. POWERS, Representations of positive polynomials on non-compact semialgebraic sets via KKT ideals, J. Pure Appl. Algebra, 209 (2007), pp. 189-200.
10. A. EINSTEIN, B. PODOLSKY, AND N. ROSEN, Can quantum-mechanical description of physical reality be considered complete?, Phys. Rev., 47 (1935), pp. 777-780.
11. L. FAYBUSOVICH, Global optimization of homogeneous polynomials on the simplex and on the sphere, in Frontiers in Global Optimization, C. Floudas and P. Pardalos, eds., Kluwer Academic Publishers, Norwell, MA, 2003.
12. K. FUJISAWA, Y. FUTAKATA, M. KOJIMA, S. MATSUYAMA, S. NAKAMURA, K. NAKATA, AND M. YAMASHITA, SDPA-M (SemiDefinite Programming Algorithm in MATLAB), http://homepage.mac.com/klabtitech/sdpa-homepage/download.html.
13. M. FU, Z.-Q. LUO, AND Y. YE, Approximation algorithms for quadratic programming, J. Comb. Optim., 2 (1998), pp. 29-50.
14. V. S. GRIGORASCU AND P. A. REGALIA, Tensor displacement structures and polyspectral matching, in Fast Reliable Algorithms for Structured Matrices, T. Kailath and A. H. Sayed, eds., SIAM, Philadelphia, 1999.
15. L. GURVITS, Classical deterministic complexity of Edmonds' problem and quantum entanglement, in Proceedings of the Thirty-Fifth ACM Symposium on Theory of Computing 10-19, New York, 2003.
16. D. HAN, H. H. DAI, AND L. QI, Conditions for strong ellipticity of anisotropic elastic materials, J. Elasticity, 97 (2009), pp. 1-13.
17. S. HE, Z-Q. LUO, J. NIE, and S. ZHANG, Semidefinite relaxation bounds for indefinite homogeneous quadratic optimization, SIAM J. Optim., 19 (2008), pp. 503-523.
18. D. JIBETEAN AND M. LAURENT, Semidefinite approximations for global unconstrained polynomial optimization, SIAM J. Optim., 16 (2005), pp. 490-514.
19. E. KOFIDIS AND P. A. REGALIA, On the best rank-1 approximation of higher-order supersymmetric tensors, SIAM J. Matrix Anal. Appl., 23 (2002), pp. 863-884.
20. J. LASSERRE, Global optimization with polynomials and the problem of moments, SIAM J. Optim., 11 (2001), pp. 796-817.
21. 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.
22. Z.-Q. LUO AND S. Z. ZHANG, A Semidefinite Relaxation Scheme for Multivariate Quartic Polynomial Optimization with Quadratic Constraints, manuscript, Department of SEEM, Chinese University of Hong Kong, Ma Liu Shui, Sha Tin, New Territories, Hong Kong, 2009.
23. T. S. MOTZKIN AND E. G. STRAUS, Maxima for graphs and a new proof of a theorem of Túran, Canad. J. Math., 17 (1965), pp. 533-540.
24. J. NIE, J. DEMMEL, AND B. STURMFELS, Minimizing polynomials via sum of squares over the gradient ideal, Math. Program., 106 (2006), pp. 587-606.
25. J. NIE AND M. SCHWEIGHOFER, On the complexity of putinar's positivstellensatz, J. Complexity, 23 (2007), pp. 135-150.
26. C. L. NIKIAS AND A. P. PETROPULU, Higher-Order Spectra Analysis, A Nonlinear Signal Processing Framework, Prentice-Hall, Englewood Cliffs, NJ, 1993.
27. P. PARRILO, Semidefinite programming relaxations for semialgebraic problems, Math. Program. Ser. B, 96 (2003), pp. 293-320.
28. L. QI, Eigenvalues of a real supersymmetric tensor, J. Symbolic Comput., 40 (2005), pp. 1302-1324.
29. L. QI, H. H. DAI, AND D. HAN, Conditions for Strong Ellipticity and M-eigenvalues, Front. Math. China 4, 2009, pp. 349-364.
30. L. QI, F. WANG, AND Y. WANG, Z-Eigenvalue methods for a global optimization polynomial optimization problem, Math. Program., 118 (2009), pp. 301-306.
31. B. REZNICK, Some concrete aspects of Hilbert's 17th problem, in Contemp. Math. 253, American Mathematical Society, Providence, RI, 2000, pp. 251-272
32. P. ROSAKIS, Ellipticity and deformations with discontinuous deformation gradients in finite elastostatics, Arch. Ration. Mech. Anal., 109 (1990), pp. 1-37.
33. E. SCHRÖDINGER, Die gegenwärtige situation in der quantenmechanik, Naturwissenschaften, 23 (1935), pp. 807-812, 823-828, 844-849.
34. H. C. SIMPSON AND S. J. SPECTOR, On copositive matrices and strong ellipticity for isotropic elastic materials, Arch. Ration. Mech. Anal., 84 (1983), pp. 55-68.
35. J. F. STURM AND S. Z. ZHANG, On cones of nonnegative quadratic functions, Math. Oper. Res., 28 (2003), pp. 246-267.
36. P. TSENG, Further results on approximating nonconvex quadratic optimization by semidefinite programming re laxation, SIAM J. Optim., 14 (2003), pp. 268-283.
37. Y. YE, On affine scaling algorithms for nonconvex quadratic programming, Math. Program., 56 (1992), pp. 285-300.
38. Y. YE AND S. Z. ZHANG, New results on quadratic minimization, SIAM J. Optim., 14 (2003), pp. 245-267.
39. Y. WANG AND M. ARON, A reformulation of the strong ellipticity conditions for unconstrained hyperelastic media, J. Elasticity, 44 (1996), pp. 89-96.
40. T. ZHANG AND G. H. GOLUB, Rank-one approximation of higher-order tensors, SIAM J. Matrix Anal. Appl., 23 (2001), pp. 534-550.