Malick, Jérôme A dual approach to semidefinite least-squares problems. (English) Zbl 1080.65027 SIAM J. Matrix Anal. Appl. 26, No. 1, 272-284 (2004). In an Euclidean space, a projection is studied onto the intersection of an affine subspace and a closed convex set. A Lagrangian dualization of this least-squares problem is proposed. This leads to a convex differentiable problem, which can be solved with a quasi-Newton algorithm. The results are applied to the cone of positive semidefinite matrices.Such projection problems arise in portfolio risk analysis, see N. J. Higham [IMA J. Numer. Anal. 22, No. 3, 329–343 (2002; Zbl 1006.65036)], and in robust quadratic optimization, see P. I. Davies and N. J. Higham [BIT 40, No. 4, 640–651 (2000; Zbl 0969.65036)]. Numerical experiments show that fairly large problems can be solved efficiently. Reviewer: Oleksandr Kukush (Kyïv) Cited in 1 ReviewCited in 52 Documents MSC: 65F20 Numerical solutions to overdetermined systems, pseudoinverses 65K05 Numerical mathematical programming methods 90C22 Semidefinite programming 91G60 Numerical methods (including Monte Carlo methods) 90C53 Methods of quasi-Newton type Keywords:Lagrangian duality; semidefinite optimization; calibration of covariance matrices; least-squares problem; quasi-Newton algorithm; numerical experiments Citations:Zbl 1006.65036; Zbl 0969.65036 Software:SeDuMi; PLCP PDFBibTeX XMLCite \textit{J. Malick}, SIAM J. Matrix Anal. Appl. 26, No. 1, 272--284 (2004; Zbl 1080.65027) Full Text: DOI