Regularized Newton methods for convex minimization problems with singular solutions. (English) Zbl 1056.90111

Li, Dong-Hui; Fukushima, Masao; Qi, Liqun; Yamashita, Nobuo

Comput. Optim. Appl. 28, No. 2, 131-147 (2004).

Summary: This paper studies convergence properties of regularized Newton methods

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for minimizing a convex function whose Hessian matrix

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may be singular everywhere. We show that if the objective function is LC\(_2\), then the methods possess local quadratic convergence under a local error bound condition without the requirement of isolated nonsingular solutions. By using a backtracking line search, we globalize an inexact regularized Newton method. We show that the unit stepsize is accepted eventually. Limited numerical experiments are presented, which show the practical advantage of the method.

MSC:

90C25	Convex programming
90C53	Methods of quasi-Newton type

Keywords:

minimization problem; regularized Newton methods; global convergence; quadratic convergence; unit step