We found a match
Your institution may have rights to this item. Sign in to continue.
- Title
Convergence Properties of the Regularized Newton Method for the Unconstrained Nonconvex Optimization.
- Authors
Ueda, Kenji; Yamashita, Nobuo
- Abstract
The regularized Newton method (RNM) is one of the efficient solution methods for the unconstrained convex optimization. It is well-known that the RNM has good convergence properties as compared to the steepest descent method and the pure Newton’s method. For example, Li, Fukushima, Qi and Yamashita showed that the RNM has a quadratic rate of convergence under the local error bound condition. Recently, Polyak showed that the global complexity bound of the RNM, which is the first iteration k such that | ∇ f( x k)|≤ ε, is O( ε−4), where f is the objective function and ε is a given positive constant. In this paper, we consider a RNM extended to the unconstrained “nonconvex” optimization. We show that the extended RNM (E-RNM) has the following properties. (a) The E-RNM has a global convergence property under appropriate conditions. (b) The global complexity bound of the E-RNM is O( ε−2) if ∇2 f is Lipschitz continuous on a certain compact set. (c) The E-RNM has a superlinear rate of convergence under the local error bound condition.
- Subjects
NEWTON-Raphson method; CONVEX functions; STOCHASTIC convergence; METHOD of steepest descent (Numerical analysis); LIPSCHITZ spaces; MATHEMATICAL optimization
- Publication
Applied Mathematics & Optimization, 2010, Vol 62, Issue 1, p27
- ISSN
0095-4616
- Publication type
Article
- DOI
10.1007/s00245-009-9094-9