Robust solutions to least-squares problems with uncertain data

SIAM Journal on Matrix Analysis and Applications

Volumn 18, Issue 4, 1997, Pages 1035-1064

  El Ghaoui, Laurent ^a   Lebret, Hervé ^a  

^a ENSTA PARISTECH   (France)

Author keywords

Ill conditioned problem; Least squares problems; Regularization; Robust identification; Robust interpolation; Robustness; Second order cone programming; Semidefinite programming; Uncertainty

Indexed keywords

EID: 0031536271     PISSN: 08954798     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0895479896298130     Document Type: Article

References (48)

1. K. D. ANDERSEN, An efficient Newton barrier method for minimizing a sum of Euclidean norms, SIAM J. Optim., 6 (1996), pp. 74-95.
2. A. BJÖRCK, Component-wise perturbation analysis and error bounds for linear least squares solutions, BIT, 31 (1991), pp. 238-244.
3. J. F. BONNANS. R. COMINETTI, AND A. SHAPIRO, Sensitivity analysis of optimization problems under abstract constraints, SIAM J. Optim., submitted.
4. S. BOYD, L. EI. GHAOUI, E. FERON, AND V. BALAKRISHNAN, Linear Matrix Inequalities in System and Control Theory, in Studies in Applied Mathematics, SIAM, Philadelphia, PA, 1994.
5. S. CHANDRASEKARAN, G. H. GOLUB, M. GU, AND A. H. SAYED, A new linear least-squares type model for parameter estimation in the presence of data uncertainties, SIAM J. Matrix Anal. Appl., submitted.
6. G. E. COXSON AND C. L. DEMARCO, Computing the Real Structured Singular Value is NP-Hard, Tech. report ECE-92-4, Dept. of Elec. and Comp. Eng., University of Wisconsin-Madison, Madison, WI, June 1992.
7. 2 approximation problems, Linear Algebra Appl., 188-189 (1993), pp. 163-207.
8. G. DEMOMENT, Image reconstruction and restoration: Overview of common estimation problems, IEEE Trans. Acoustic Speech and Signal Processing, 37 (1989). pp. 2024-2036.
9. J. DOYLE, M. NEWLIN, F. PAGANINI, AND J. TIERNO, Unifying robustness analysis and system ID, in Proc. IEEE Conf. on Decision and Control, December 1994, pp. 3667-3672.
10. L. EL GHAOUI, R. NIKOUKHAH, AND F. DELEBECQUE, LMITOOL: A Front-End for LMI Optimization, User's Guide, February 1995. Available via anonymous ftp from ftp.ensta.fr/pub/elghaoui/lmitool.
11. L. ELDEN, Algorithms for the regularization of ill conditioned least-squares problems, BIT, 17 (1977), pp. 134-145.
12. L. ELDEN, Perturbation theory for the least-squares problem with linear equality constraints, BIT, 24 (1985), pp. 472-476.
13. M. K. H. FAN, A. L. TITS, AND J. C. DOYLE, Robustness in the presence of mixed parametric uncertainty and unmodeled dynamics, IEEE Trans. Automat. Control, 36 (1991), pp. 25-38.
14. R. D. FIERRO AND J. R. BUNCH, Collinearity and total least squares, SIAM J. Matrix Anal. Appl., 15 (1994), pp. 1167-1181.
15. M. FURUYA, H. OHMORI, AND A. SANO, Optimization of weighting constant for regularization in least squares system identification, Trans. Inst. Elec. Inform. Comm. Eng. A, J72A (1989), pp. 1012-1015.
16. M. X. GOEMANS AND D. P. WILLIAMSON, .878-approximation for MAX CUT and MAX 2SAT, in Proc. 26th ACM Symp. Theor. Computing, 1994, pp. 422-431.
17. G. H. GOLUB AND C. F. VAN LOAN, An analysis of the total least squares problem, SIAM J. Numer. Anal., 17 (1980), pp. 883-893.
18. G. H. GOLUB AND C. F. VAN LOAN, Matrix Computations, 2nd ed., Johns Hopkins University Press, Baltimore, MD, 1989.
19. G. H. GOLUB AND U. VON MATT, Quadratically constrained least squares and quadratic problems, Numer. Math., 59 (1991), pp. 561-580.
20. M. L. HAMBABA, The robust generalized least-squares estimator, Signal Processing, 26 (1992), pp. 359-368.
21. M. HANKE AND P. C. HANSEN, Regularization methods for large-scale problems, Surveys on Mathematics for Industry, 3 (1993), pp. 253-315.
22. D. J. HIGHAM AND N. J. HIGHAM, Backward error and condition of structured linear systems, SIAM J. Matrix Anal. Appl., 13 (1992), pp. 162-175.
23. B. R. HUNT, The application of constrained least-squares estimation to image restoration by digital computer, IEEE Trans Comput., C-22 (1973), pp. 805-812.
24. T. IWASAKI AND R. E. SKELTON, All controllers for the general H∞ control problem: LMI existence conditions and state space formulas, Automatica, 30 (1994), pp. 1307-1317.
25. C. JACQUEMONT, Error-in-Variables Robust Least-Squares, Tech. rep., Ecole Nat. Sup. Techniques Avancies, 32, Bd. Victor, 75739 Paris, France, December 1995.
26. G. LADY AND J. MAYBEE, Qualitatively invertible matrices, J. Math. Social Sciences, 6 (1983), pp. 397-407.
27. H. LEBRET, Synthèse de diagrammes de réseaux d'antennes par optimisation convexe, Ph.D. thesis, UFR Structure et Propriétés de la Matière, mention Electronique, Université de \ Rennes I, Prance, 1994.
28. H. LEBRET, Antenna pattern synthesis through convex optimization, in Advanced Signal Processing Algorithms, Proc. SPIE 2563, F. T. Luk, ed., 1995, pp. 182-192.
29. L. LEE AND A. TITS, On continuity/discontinuity in robustness indicators, IEEE Trans. Automat. Control, 38 (1993), pp. 1551-1553.
30. K. MILLER, Least squares methods for ill-posed problems with a prescribed bound. SIAM J. Math. Anal., 1 (1970), pp. 52-74.
31. M. Z. NASHED, Operator-theoretic and computational approaches to ill-posed problems with applications to antenna theory, IEEE Trans. Antennas and Propagation, 29 (1981), pp. 220-231.
32. A. NEMIROVSKY, Several NP-hard problems arising in robust stability analysis, Mathematics of Control, Signals, and Systems, 6 (1993), pp. 99-105.
33. Y. NESTEROV AND A. NEMIROVSKY, Interior Point Polynomial Methods in Convex Programming: Theory and Applications, SIAM. Philadelphia, PA, 1994.
34. J. NORTON, Identification and application of bounded parameter models, Autornatica, 31 (1987), pp. 497-507.
35. F. OUSTRY, L. EL GHAOUI, AND H. LEBRET. Robust solutions to uncertain semidefinite programs, SIAM J. Optim., submitted, 1996.
36. C. PAPADIMITRIOU AND M. YANNAKAKIS, Optimization, approximation and complexity classes, J. Comput. System Sci., 43 (1991), pp. 425-440.
37. J. R. PARTINGTON AND P. M. MÄKILÄ, Worst-case analysis of the least-squares method and related identification methods, Systems Control Lett., 24 (1995), pp. 193-200.
38. S. POLJAK AND J. ROHN, Checking robust nonsingulanty is NP-hard, Math. Control Signals Systems, 6 (1993), pp. 1-9.
39. B. L. SHADER, Least squares sign-solvability. SIAM J. Matrix Anal. Appl., 16 (1995), pp. 1056-1073.
40. R. SMITH AND J. DOYLE, Model validation: A connection between robust control and identification, IEEE Trans. Automat. Control, 37 (1992), pp. 942-952.
41. R. J. STERN AND H. WOLKOWICZ, Indefinite trust region subproblems and nonsymmetric eigenvalue perturbations, SIAM J. Optim., 5 (1995), pp. 286-313.
42. R. TEMPO, Worst-case optimality of smoothing algorithms for parametric system identification, Automatica, 31 (1995), pp. 759-764.
43. A. TIKHONOV AND V. ARSENIN, Solutions of Ill-Posed Problems, Wiley, New York, 1977.
44. S. VAN HUFFEL AND J. VANDEWALLE, The total least squares problem: Computational aspects and analysis, in Frontiers in Applied Mathetics 9, SIAM, Philadelphia, PA, 1991.
45. L. VANDENBERGHE AND S. BOYD, SP, Software for Semidefinite Programming, User's Guide, Dec. 1994. Available via anonymous ftp from isl.stanford.edu under /pub/boyd/semidef_prog.
46. L. VANDENBERGHE AND S. BOYD, Semidefinite programming, SIAM Rev., 38 (1996), pp. 49-95.
47. M. E. ZERVAKIS AND T. M. KWON, Robust estimation techniques in regularized image restoration, Op. Eng., 31 (1992), pp. 2174-2190.
48. K. ZHOU, J. DOYLE, AND K. GLOVER, Robust and Optimal Control, Prentice-Hall, Englewood Cliffs, NJ, 1995.