Maximum block improvement and polynomial optimization

SIAM Journal on Optimization

Volumn 22, Issue 1, 2012, Pages 87-107

  Chen, Bilian ^a   He, Simai ^b   Li, Zhening ^c   Zhang, Shuzhong ^d  

^a CHINESE UNIVERSITY OF HONG KONG   (Hong Kong)

^b CITY UNIVERSITY OF HONG KONG   (Hong Kong)

^c SHANGHAI UNIVERSITY   (China)

^d UNIVERSITY OF MINNESOTA   (United States)

Author keywords

Block coordinate descent; Polynomial optimization problem; Tensor form

Indexed keywords

COMPUTATIONAL COSTS; COORDINATE DESCENT; GLOBAL OPTIMAL SOLUTIONS; HOMOGENEOUS POLYNOMIALS; KKT POINTS; NON-LINEAR OPTIMIZATION; NUMERICAL EXPERIMENTS; POLYNOMIAL OPTIMIZATION; POLYNOMIAL OPTIMIZATION PROBLEM; RELAXATION PROBLEM; SEARCH METHOD; STATIONARY POINTS; STATIONARY SOLUTIONS; TEST INSTANCES;

CONSTRAINED OPTIMIZATION; ECHO SUPPRESSION; NUMERICAL METHODS; SPHERES; TENSORS;

POLYNOMIALS;

EID: 84861636156     PISSN: 10526234     EISSN: None     Source Type: Journal    
DOI: 10.1137/110834524     Document Type: Article

References (65)

1. W. Ai, Y. Huang, and S. Zhang, New results on Hermitian matrix rank-one decomposition. Math. Program. Ser. A, 128 (2011), pp. 253-283.
2. M. F. Anjos and J. B. Lasserre, Handbook of Semidefinite, Conic and Polynomial Optimization, Springer, New York, 2011.
3. B. W. Bader and T. G. Kolda, Matlab Tensor Toolbox Version 2.4, http://csmr.ca. sandia.gov/tgkolda/TensorToolbox (2010).
4. A. Barmpoutis, B. Jian, B. C. Vemuri, and T. M. Shepherd, Symmetric Positive 4th Order Tensors and Their Estimation from Diffusion Weighted MRI, Lecture Notes in Comput. Sci. 4584, N. Karssemijer and B. Lelieveldt, eds., Springer, New York, 2007, pp. 308-319.
5. D. P. Bertsekas, Nonlinear Programming, 2nd ed., Athena Scientific, Belmont, MA, 1999.
6. J. Bochnak and J. Siciak, Polynomials and multilinear mappings in topological vector spaces, Stud. Math., 39 (1971), pp. 59-76.
7. P. H. Calamai and J. J. Moré, Projected gradient methods for linearly constrained problems, Math. Program., 39 (1987), pp. 93-116.
8. J. D. Carroll and J. J. Chang, Analysis of individual differences in multidimensional scaling via an N-way generalization of "Eckart-Young" decomposition, Psychometrika, 35 (1970), pp. 283-319.
9. 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.
10. E. F. Gonzalez and Y. Zhang, Accelerating the Lee-Seung Algorithm for Nonnegative Matrix Factorization, preprint, Department of Computational and Applied Mathematics, Rice University, March 2005.
11. L. Grippo and M. Sciandrone, On the convergence of the block nonlinear Gauss-Seidel method under convex constraints, Oper. Res. Lett., 26 (2000), pp. 127-136.
12. E. de Klerk, The complexity of optimizing over a simplex, hypercube or sphere: A short survey, Cent. Eur. J. Oper. Res., 16 (2008), pp. 111-125.
13. E. de Klerk, M. Laurent, and P. A. Parrilo, A PTAS for the minimization of polynomials of fixed degree over the simplex, Theoret. Comput. Sci., 261 (2006), pp. 210-225.
14. L. De Lathauwer, B. De Moor, and J. Vandewalle, On the best rank-1 and rank-(R1,R2, ⋯ ,RN) approximation of higher-order tensors, SIAM J. Matrix Anal. Appl., 21 (2000), pp. 1324-1342.
15. L. De Lathauwer and D. Nion, Decompositions of a higher-order tensor in block terms-Part III: Alternating least squares algorithms, SIAM J. Matrix Anal. Appl., 30 (2008), pp. 1067-1083.
16. A. Ghosh, E. Tsigaridas, M. Descoteaux, P. Comon, B. Mourrain, and R. Deriche, A polynomial based approach to extract the maxima of an antipodally symmetric spherical function and its application to extract fiber directions from the orientation distribution function in diffusion MRI, in Proceedings of the Computational Diffusion MRI Workshop (CDMRI'08), New York, 2008.
17. M. Grant and S. Boyd, CVX: Matlab Software for Disciplined Convex Programming, version 1.2, http://cvxr.com/cvx (2010).
18. L. Gurvits, Classical deterministic complexity of Edmonds' problem and quantum entanglement, in Proceedings of the 35th ACM Symposium on Theory of Computing, New York, 2003.
19. R. A. Harshman, Foundations of the PARAFAC procedure: Models and conditions for an "explanatory" multi-modal factor analysis, UCLA Working Papers in Phonetics, 16 (1970), pp. 1-84; also available online from http://publish.uwo.ca/harshman/wpppfac0.pdf.
20. S. He, M. Li, S. Zhang, and Z.-Q. Luo, A nonconvergent example for the iterative waterfilling algorithm, Numer. Algebra Control Optim., 1 (2011), pp. 147-150.
21. S. He, Z. Li, and S. Zhang, Approximation algorithms for homogeneous polynomial optimization with quadratic constraints, Math. Program. Ser. B, 125 (2010), pp. 353-383.
22. S. He, Z. Li, and S. Zhang, General Constrained Polynomial Optimization: An Approximation Approach, Technical report SEEM2009-06, Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, 2009.
23. S. He, Z. Li, and S. Zhang, Approximation Algorithms for Discrete Polynomial Optimization, Technical report SEEM2010-01, Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, 2010.
24. D. Henrion and J. B. Lasserre, GloptiPoly: Global optimization over polynomials with Matlab and SeDuMi, ACM Trans. Math. Software, 29 (2003), pp. 165-194.
25. D. Henrion, J. B. Lasserre, and J. Loefberg, GloptiPoly 3: Moments, optimization and semidefinite programming, Optim. Methods Softw., 24 (2009), pp. 761-779.
26. C. J. Hillar and L.-H. Lim, Most Tensor Problems are NP Hard, preprint; also available online from http://www.stat.uchicago.edu/lekheng/work/np.pdf (2010).
27. J.-B. Hiriart-Urruty, A new series of conjectures and open questions in optimization and matrix analysis, ESAIM Control Optim. Calc. Var., 15 (2009), pp. 454-470.
28. P. M. J. Hof, C. Scherer, and P. S. C. Heuberger, Model-Based Control: Bridging Rigorous Theory and Advanced Technology, Part 1, Springer, New York, 2009, pp. 49-68.
29. S. Khot and A. Naor, Linear equations modulo 2 and the L1 diameter of convex bodies, in Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science, 2007, pp. 318-328.
30. 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.
31. T. G. Kolda and B. W. Bader, Tensor decompositions and applications, SIAM Rev., 51 (2009), pp. 455-500.
32. J. B. Lasserre, Global optimization with polynomials and the problem of moments, SIAM J. Optim., 11 (2001), pp. 796-817.
33. J. B. Lasserre, Polynomials nonnegative on a grid and discrete representations, Trans. Amer. Math. Soc., 354 (2001), pp. 631-649.
34. J. B. Lasserre, Convergent SDP relaxations in polynomial optimization with sparsity, SIAM J. Optim., 17 (2006), pp. 822-843.
35. J. B. Lasserre, A sum of squares approximation of nonnegative polynomials, SIAM Rev., 49 (2007), pp. 651-669.
36. Z. Li, Polynomial Optimization Problems-Approximation Algorithms and Applications, Ph.D. thesis, The Chinese University of Hong Kong, 2011.
37. C. Ling, J. Nie, L. Qi, and Y. Ye, Biquadratic optimization over unit spheres and semidefinite programming relaxations, SIAM J. Optim., 20 (2009), pp. 1286-1310.
38. Z.-Q. Luo and J.-S. Pang, Analysis of iterative waterfilling algorithm for multiuser power control in digital subscriber lines, EURASIP J. Appl. Signal Process., 2006 (2006), pp. 1-10.
39. Z.-Q. Luo and S. Zhang, A semidefinite relaxation scheme for multivariate quartic polynomial optimization with quadratic constraints, SIAM J. Optim., 20 (2010), pp. 1716-1736.
40. B. Mourrain and J. P. Pavone, Subdivision methods for solving polynomial equations, J. Symbolic Comput., 44 (2009), pp. 292-306.
41. B. Mourrain and P. Trébuchet, Generalized normal forms and polynomial system solving, in Proceedings of the International Symposium on Symbolic and Algebraic Computation, M. Kauers, ed., 2005, ACM Press, New York, pp. 253-260.
42. Y. Nesterov, Random Walk in a Simplex and Quadratic Optimization over Convex Polytopes, CORE Discussion Paper, UCL, Louvain-la-Neuve, Belgium, 2003.
43. Y. Nesterov and A. Nemirovski, Interior-Point Polynomial Algorithms in Convex Programming, SIAM Stud. Appl. Math., 1994.
44. Network-Enabled Optimization Systems (NEOS) Server for Optimization, http://www.neosserver. org/neos.
45. G. Ni and Y. Wang, On the best rank-1 approximation to higher-order symmetric tensors, Math. Comput. Model., 46 (2007), pp. 1345-1352.
46. D. Nion and L. De Lathauwer, An enhanced line search scheme for complex-valued tensor decompositions, Appl. DS-CDMA Signal Process., 88 (2008), pp. 749-755.
47. P. A. Parrilo, Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimization, Ph.D. dissertation, California Institute of Technology, 2000.
48. P. A. Parrilo, Semidefinite programming relaxations for semialgebraic problems, Math. Program. Ser. B, 96 (2003), pp. 293-320.
49. S. Prajna, A. Papachristodoulou, and P. A. Parrilo, Introducing SOSTOOLS: A general purpose sum of squares programming solver, in Proceedings of the 41st IEEE Conference on Decision and Control, Las Vegas, 2002.
50. L. Qi, Eigenvalues of a real supersymmetric tensor, J. Symbolic Comput., 40 (2005), pp. 1302-1324.
51. M. J. D. Powell, On search directions for minimization algorithms, Math. Program., 4 (1973), pp. 193-201.
52. L. Qi, Eigenvalues and invariants of tensors, J. Math. Anal. Appl., 325 (2007), pp. 1363-1377.
53. L. Qi and K. L. Teo, Multivariate polynomial minimization and its application in signal processing, J. Global Optim., 26 (2003), pp. 419-433.
54. L. Qi, F. Wang, and Y. Wang, Z-eigenvalue methods for a global polynomial optimization problem, Math. Program. Ser. A, 118 (2009), pp. 301-316.
55. M. Rajih, P. Comon, and R. A. Harshman, Enhanced line search: A novel method to accelerate PARAFAC, SIAM J. Matrix Anal. Appl., 30 (2008), pp. 1128-1147.
56. A. M.-C. So, Deterministic approximation algorithms for sphere constrained homogeneous polynomial optimization problems, Math. Program. Ser. B, 129 (2011), pp. 357-382.
57. S. Soare, J. W. Yoon, and O. Cazacu, On the use of homogeneous polynomials to develop anisotropic yield functions with applications to sheet forming, Int. J. Plasticity, 24 (2008), pp. 915-944.
58. J. F. Sturm, SeDuMi 1.02, a Matlab toolbox for optimization over symmetric cones, Optim. Methods Softw., 11/12 (1999), pp. 625-653.
59. J. F. Sturm and S. Zhang, On cones of nonnegative quadratic functions, Math. Oper. Res., 28 (2003), pp. 246-267.
60. P. Tseng, Convergence of a block coordinate descent method for nondifferentiable minimization, J. Optim. Theory Appl., 109 (2001), pp. 475-494.
61. H.Waki, S. Kim, M. Kojima, M. Muramatsu, and H. Sugimoto, Algorithm 883: SparsePOP: A sparse semidefinite programming relaxation of polynomial optimization problems, ACM Trans. Math. Software, 35 (2008), pp. 1-13.
62. Y. Wang and L. Qi, On the successive supersymmetric rank-1 decomposition of higher-order supersymmetric tensors, Numer. Linear Algebra Appl., 14 (2007), pp. 503-519.
63. S. Wright, Accelerated Block-Coordinate Relaxation for Regularized Optimization, SIAM J. Optim., to appear.
64. Y. Ye and S. Zhang, New results on quadratic minimization, SIAM J. Optim., 14 (2003), pp. 245-267.
65. T. Zhang and G. H. Golub, Rank-one approximation to high order tensors, SIAM J. Matrix Anal. Appl., 23 (2001), pp. 534-550.