A dual approach to semidefinite least-squares problems.

*(English)*Zbl 1080.65027In 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)

##### MSC:

65F20 | Overdetermined systems, pseudoinverses (numerical linear algebra) |

65K05 | Mathematical programming (numerical methods) |

90C22 | Semidefinite programming |

91B28 | Finance etc. (MSC2000) |

90C53 | Methods of quasi-Newton type |