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A dual approach to semidefinite least-squares problems. (English) Zbl 1080.65027

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.

65F20Overdetermined systems, pseudoinverses (numerical linear algebra)
65K05Mathematical programming (numerical methods)
90C22Semidefinite programming
91B28Finance etc. (MSC2000)
90C53Methods of quasi-Newton type