# zbMATH — the first resource for mathematics

A projected semismooth Newton method for problems of calibrating least squares covariance matrix. (English) Zbl 1218.90220
Summary: We propose a projected semismooth Newton method to solve the problem of calibrating least squares covariance matrix with equality and inequality constraints. The method is globally and quadratically convergent with proper assumptions. The numerical results show that the proposed method is efficient and comparable with existing methods.

##### MSC:
 90C53 Methods of quasi-Newton type 90C22 Semidefinite programming 90C20 Quadratic programming
Full Text:
##### References:
 [1] Alizadeh, F.; Haeberly, J.-P.A.; Overton, M.L., Complementarity and nondegenracy in semidefinite programming, Mathematical programming, 77, 111-128, (1997) · Zbl 0890.90141 [2] Bertsekas, D.P., Projected Newton methods for optimization problems with simple constraints, SIAM journal on control and optimization, 20, 221-246, (1982) · Zbl 0507.49018 [3] Bonnans, J.F.; Shapiro, A., Perturbation analysis of optimization problems, (2000), Springer · Zbl 0966.49001 [4] Borsdorf, R.; Higham, N.J., A preconditioned Newton algorithm for the nearest correlation matrix, IMA journal of numerical analysis, 30, 94-107, (2010) · Zbl 1188.65055 [5] Boyd, S.; Xiao, L., Least-squares covariance matrix adjustment, SIAM journal on matrix analysis and applications, 27, 532-546, (2005) · Zbl 1099.65039 [6] Chan, Z.X.; Sun, D., Constraint nondegeneracy, strong regularity and nonsingularity in semidefinite programming, SIAM journal on optimization, 19, 370-396, (2008) · Zbl 1190.90116 [7] Chen, Y.D.; Gao, Y.; Liu, Y.-J., An inexact SQP Newton method for convex $$S C^1$$ minimization problems, Journal of optimization theory and applications, 146, 33-49, (2010) · Zbl 1197.90314 [8] Facchinei, F.; Fischer, A.; Kanzow, C., On the accurate identification of active constraints, SIAM journal on optimization, 9, 14-32, (1998) · Zbl 0960.90080 [9] Gao, Y.; Sun, D.F., Calibrating least squares covariance matrix problems with equality and inequality constraints, SIAM journal on matrix analysis and applications, 31, 1432-1457, (2009) · Zbl 1201.49031 [10] Higham, N.J., Computing the nearest correlation matrix—a problem from finance, IMA journal of numerical analysis, 22, 329-343, (2002) · Zbl 1006.65036 [11] Kanzow, C.; Qi, H.-D., A QP-free constrained Newton-type method for variational inequality problems, Mathematical programming, 85, 81-106, (1999) · Zbl 0958.65078 [12] Malick, J., A dual approach to semidefinite least-squares problems, SIAM journal on matrix analysis and applications, 26, 272-284, (2004) · Zbl 1080.65027 [13] Paige, C.C.; Saunders, M.A., Solution of sparse indefinite systems of linear equations, SIAM journal on numerical analysis, 12, 617-629, (1975) · Zbl 0319.65025 [14] Qi, H.-D.; Sun, D.F., A quadratically convergent Newton method for computing the nearest correlation matrix, SIAM journal on matrix analysis and applications, 28, 360-385, (2006) · Zbl 1120.65049 [15] Qi, H.-D.; Sun, D.F., Correlation stress testing for value-at-risk: an unconstrained convex optimization approach, Computational optimization and applications, 45, 427-462, (2010) · Zbl 1198.91091 [16] Qi, H.-D., Positive semidefinite matrix completions on chordal graphs and constraint nondegeneracy in semidefinite programming, Linear algebra and its applications, 430, 1151-1164, (2009) · Zbl 1226.90070 [17] Sun, D.F., The strong second-order sufficient condition and constraint nondegeneracy in nonlinear semidefinite programming and their implications, Mathematics of operations research, 31, 761-776, (2006) · Zbl 1278.90304 [18] Sun, D.F.; Sun, J., Semismooth matrix valued functions, Mathematics of operations research, 27, 150-169, (2002) · Zbl 1082.49501 [19] Zhao, X.Y.; Sun, D.F.; Toh, K.C., A newton-CG augmented Lagrangian method for semidefinite programming, SIAM journal on optimization, 20, 1737-1765, (2010) · Zbl 1213.90175
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.