A quadratically convergent newton method for computing the nearest correlation matrix

SIAM Journal on Matrix Analysis and Applications

Volumn 28, Issue 2, 2006, Pages 360-385

  Qi, Houduo ^a   Sun, Defeng ^b  

^a UNIVERSITY OF SOUTHAMPTON   (United Kingdom)

^b NATIONAL UNIVERSITY OF SINGAPORE   (Singapore)

Author keywords

Correlation matrix; Newton method; Quadratic convergence; Semismooth matrix equation

Indexed keywords

CONVERGENCE OF NUMERICAL METHODS; CORRELATION METHODS; MATRIX ALGEBRA; PROBLEM SOLVING; QUADRATIC PROGRAMMING;

CORRELATION MATRIX; QUADRATIC CONVERGENCE; SEMISMOOTH MATRIX EQUATIONS;

NEWTON-RAPHSON METHOD;

EID: 34247364678     PISSN: 08954798     EISSN: 10957162     Source Type: Journal    
DOI: 10.1137/050624509     Document Type: Article

References (50)

1. L.-E. ANDERSSON AND T. ELFVING, An algorithm for constrained interpolation, SIAM J. Sci. Statist. Comput., 8 (1987), pp. 1012-1025.
2. H. H. BAUSCHKE, J. M. BORWEIN, AND W. LI, Strong conical hull intersection property, bounded linear regularity, Jameson's property (G), and error bounds in convex optimization, Math. Program., 86 (1999), pp. 135-160.
3. H. H. BAUSCHKE, J. M, BORWEIN, AND P. TSENG, Bounded linear regularity, strong CHIP, and CHIP are distinct properties, J. Convex Anal., 7 (2000), pp. 395-412.
4. R. BHATIA, Matrix Analysis, Springer-Verlag, New York, 1997.
5. J. BORWEIN AND A. S. LEWIS, Partially finite convex programming I: Quasi relative interiors and duality theory, Math. Programming, 57 (1992), pp. 15-48.
6. J. BORWEIN AND H. WOLKOWICZ, A simple constraint qualification in infinite-dimensional programming, Math. Programming, 35 (1986), pp. 83-96.
7. S, BOYD AND L. XIAO, Least-squares covariance matrix adjustment, SIAM J. Matrix Anal. Appl., 27 (2005), pp. 532-546.
8. X. CHEN, H. QI, AND P. TSENG, Analysis of nonsmooth symmetric-matrix-valued functions with applications to semidefinite complementarity problems, SIAM J. Optim., 13 (2003), pp. 960-985.
9. C. K. CHUI, F. DEUTSCH, AND J. D. WARD, Constrained best approximation in Hilbert space, Constr. Approx., 6 (1990), pp. 35-64.
10. C. K. CHUI, F. DEUTSCH, AND J. D. WARD, Constrained best approximation in Hilbert space II, J. Approx. Theory, 71 (1992), pp. 213-238.
11. F. H. CLARKE, Optimization and Nonsmooth Analysis, John Wiley and Sons, New York, 1983.
12. J. E. DENNIS AND R. B. SCHNABEL, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Prentice-Hall, Englewood CHffs, NJ, 1983.
13. F. DEUTSCH, Best Approximation in Inner Product Spaces, CMS Books Math./Ouvrages Math. SMC 7, Springer-Verlag, New York, 2001.
14. F. DEUTSCH, W. LI, AND J. D. WARD, A dual approach to constrained interpolation from a convex subset of Hilbert space, J. Approx. Theory, 90 (1997), pp. 385-414.
15. F. DEUTSCH, W. LI, AND J. D. WARD, Best approximation from the intersection of a closed convex set and a polyhedron in Hilbert space, weak Slater conditions, and the strong conical hull intersection property, SIAM J. Optim., 10 (1999), pp. 252-268.
16. W. F. DONOGHUE, Monotone Matrix Functions and Analytic Continuation, Springer-Verlag, New York, 1974.
17. A. L. DONTCHEV AND B. D. KALCHEV, Duality and well-posedness in convex interpolation, Numer. Funct. Anal. Optim., 10 (1989), pp. 673-689.
18. A. L. DONTCHEV. H.-D. QI, AND L. QI, Convergence of Newton's method for convex best interpolation, Numer. Math., 87 (2001), pp. 435-456.
19. A. L. DONTCHEV, H.-D. QI, AND L. QI, Quadratic convergence of Newton's method for convex interpolation and smoothing, Constr. Approx., 19 (2003), pp. 123-143.
20. R. L. DYKSTRA, An algorithm for restricted least squares regression, J. Amer. Statist. Assoc., 78 (1983), pp. 837-842.
21. 1 functions and the Maratos effect, Oper. Res. Lett., 17 (1995), pp. 131-137.
22. J. FAVARD, Sur l'interpolation, J. Math. Pures Appl. (9), 19 (1940), pp. 281-306.
23. M. R. HESTENES AND E. STIEFEL, Methods of conjugate gradients for solving linear systems, J. Research Nat. Bur. Standards, 49 (1952), pp. 409-436.
24. N. J. HIGHAM, Computing a nearest symmetric positive semidefinite matrix, Linear Algebra Appl., 103 (1988), pp. 103-118.
25. N. J. HIGHAM, Computing the nearest correlation matrix - a problem from finance, IMA J. Numer. Anal. 22 (2002), pp. 329-343.
26. R. A. HORN AND C. R. JOHNSON, Matrix Analysis, Cambridge University Press, Cambridge, UK, 1985.
27. U. HORNUNG, Interpolation by smooth functions under restriction on the derivatives, J. Approx. Theory, 28 (1980), pp. 227-237.
28. p norm (1 < p < ∞) of the kth derivative, in Mathematics and Mathematical Education (Sunny Beach, 1984), Bulg. Akad. Nauk, Sofia, Bulgaria, 1984, pp. 31-42.
29. L. D. IRVINE, S. P. MARIN, AND P. W. SMITH, Constrained interpolation and smoothing, Constr. Approx., 2 (1986), pp. 129-151.
30. 1 equations, J. Oper. Res. Soc. Japan, 29 (1986), pp. 352-374.
31. N. KRISLOCK, J. LANG. J. VARAH, AND D. K. PAI, Local compliance estimation via positive. semidefinite constrained least squares, IEEE Transactions on Robotics and Automation, 20 (2004), pp. 1007-1011.
32. B. KUMMER, Newton's method for nondifferentiable functions, in Advances in Mathematical Optimization, Math. Res. 45, Akademie-Verlag, Berlin, 1988, pp. 114-125.
33. K. LÖWNER, Über monotone matrixfunctionen, Math. Z., 38 (1934), pp. 177-216.
34. D. G. LUENBERGER, Optimization by Vector Space Methods, John Wiley and Sons, New York, 1969.
35. J. MALICK, A dual approach to semidefinite least-squares problems, SIAM J. Matrix Anal. Appl., 26 (2004), pp. 272-284.
36. p approximation, Constr. Approx., 1 (1985), pp. 93-102.
37. C. A. MICCHELLI AND F. I. UTRERAS, Smoothing and interpolation in a convex subset of a Hilbert space, SIAM J. Sci. Statist. Comput., 9 (1988), pp. 728-746.
38. R. MIFFLIN, Semismooth and semiconvex functions in constrained optimization, SIAM J. Control Optim., 15 (1977), pp. 959-972.
39. J. NOCEDAL AND S. J. WRIGHT, Numerical Optimization, Springer-Verlag, New York, 1999.
40. J. M. ORTEGA AND W. C. RHEINBOLDT, Iterative Solution of Nonlinear Equations in Several Variables, SIAM, Philadelphia, 2000.
41. 1 minimization problems, J. Optim. Theory Appl., 85 (1995), pp. 633-648.
42. L. QL AND J. SUN, A nonsmooth version of Newton's method, Math. Programming, 58 (1993), pp. 353-367.
43. R. T. ROCKAFELLAR, Conjugate Duality and Optimization, SIAM, Philadelphia, 1974.
44. R. T. ROCKAFELLAR AND R. J.-B. WETS, Variational Analysis, Springer-Verlag, Berlin, 1998.
45. D. SUN, A further result on an implicit function theorem for locally Lipschitz functions, Oper. Res. Lett., 28 (2001), pp. 193-198.
46. D. SUN AND J. SUN, Semismooth matrix valued functions, Math. Oper. Res., 27 (2002), pp. 150-169.
47. D. SUN AND J. SUN, Löwner's Operator and Spectral Functions in Euclidean Jordan Algebras, Technical report, National University of Singapore, Singapore, 2004.
48. K. C. TOH, private communication, 2005.
49. K. C. TOH, R. H. TÜTÜNCÜ, AND M. J. TODD, Inexact Primal-Dual Path-Following Algorithms for a Special Class of Convex Quadratic SDP and Related Problems, Technical report, National University of Singapore, Singapore, 2005.
50. P. TSENG, Merit functions for semi-definite complementarity problems, Math. Programming, 83 (1998), pp. 159-185.