A smoothing Newton method for a type of inverse semi-definite quadratic programming problem

Authors:
Xiantao Xiao;Liwei Zhang;Jianzhong Zhang
Affiliations:
Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, Liaoning, China;Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, Liaoning, China;Department of Mathematics, City University of Hong Kong, Tat Chee Avenue, Hong Kong
Venue:
Journal of Computational and Applied Mathematics
Year:
2009

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Abstract

We consider an inverse problem arising from the semi-definite quadratic programming (SDQP) problem. We represent this problem as a cone-constrained minimization problem and its dual (denoted ISDQD) is a semismoothly differentiable (SC^1) convex programming problem with fewer variables than the original one. The Karush-Kuhn-Tucker conditions of the dual problem (ISDQD) can be formulated as a system of semismooth equations which involves the projection onto the cone of positive semi-definite matrices. A smoothing Newton method is given for getting a Karush-Kuhn-Tucker point of ISDQD. The proposed method needs to compute the directional derivative of the smoothing projector at the corresponding point and to solve one linear system per iteration. The quadratic convergence of the smoothing Newton method is proved under a suitable condition. Numerical experiments are reported to show that the smoothing Newton method is very effective for solving this type of inverse quadratic programming problems.