Regularized Newton Methods for Convex Minimization Problems with Singular Solutions

Authors:
Dong-Hui Li;Masao Fukushima;Liqun Qi;Nobuo Yamashita
Affiliations:
Institute of Applied Mathematics, Hunan University, Changsha, China 410082. dhli@hnu.net.cn;Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University, Kyoto 606-8501, Japan. fuku@amp.i.kyoto-u.ac.jp;Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong. maqilq@polyu.edu.hk;Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University, Kyoto 606-8501, Japan. nobuo@amp.i.kyoto-u.ac.jp
Venue:
Computational Optimization and Applications
Year:
2004

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Abstract

This paper studies convergence properties of regularized Newton methods for minimizing a convex function whose Hessian matrix may be singular everywhere. We show that if the objective function is LC2, 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.