On the convergence of a modified regularized Newton method for convex optimization with singular solutions

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

卷:239

Article

Zhou, Weijun¹  Chen, Xinlong¹ 

[1] Changsha Univ Sci & Technol, Dept Math, Changsha 410004, Hunan, Peoples R China

关键词: Convex optimization;    Newton method;    Global convergence;    Cubic convergence;   

DOI  :  10.1016/j.cam.2012.09.030

来源: Elsevier

【 摘 要 】

In this paper we propose a modified regularized Newton method for convex minimization problems whose Hessian matrices may be singular. The proposed method is proved to converge globally if the gradient and Hessian of the objective function are Lipschitz continuous. Under the local error bound condition, we first show that the method converges quadratically, which implies that parallel to x(k) - x*parallel to is equivalent to dist(x(k), X), where X is the solution set and x(k) -> x* is an element of X. Then we in turn prove the cubic convergence of the proposed method under the same local error bound condition, which is weaker than nonsingularity. (C) 2012 Elsevier B.V. All rights reserved.

【 授权许可】

Free