zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Convergence properties of the regularized Newton method for the unconstrained nonconvex optimization. (English) Zbl 1228.90087
Summary: 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 $\Vert \nabla f(x _{k })\Vert \leq \varepsilon $, is $O(\varepsilon ^{ - 4})$, where $f$ is the objective function and $\varepsilon $ 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 $O(\varepsilon ^{ - 2})$ if $\nabla ^{2} f$ is Lipschitz continuous on a certain compact set. (c) The E-RNM has a superlinear rate of convergence under the local error bound condition.

MSC:
90C26Nonconvex programming, global optimization
90C53Methods of quasi-Newton type
WorldCat.org
Full Text: DOI
References:
[1] Dan, H., Yamashita, N., Fukushima, M.: Convergence properties of the inexact Levenberg-Marquardt method under local error bound conditions. Optim. Methods Softw. 17, 605--626 (2002) · Zbl 1030.65049 · doi:10.1080/1055678021000049345
[2] Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1985) · Zbl 0576.15001
[3] Li, Y.J., Li, D.H.: Truncated regularized Newton method for convex minimizations. Comput. Optim. Appl. 43, 119--131 (2009) · Zbl 1176.90461 · doi:10.1007/s10589-007-9128-7
[4] Li, D.H., Fukushima, M., Qil, L., Yamashita, N.: Regularized Newton methods for convex minimization problems with singular solutions. Comput. Optim. Appl. 28, 131--147 (2004) · Zbl 1056.90111 · doi:10.1023/B:COAP.0000026881.96694.32
[5] Nocedal, J., Wright, S.J.: Numerical Optimization. Springer, New York (1999) · Zbl 0930.65067
[6] Polyak, R.A.: Regularized Newton method for unconstrained convex optimization. Math. Program., Ser. B 120, 125--145 (2009) · Zbl 1189.90121 · doi:10.1007/s10107-007-0143-3
[7] Yamashita, N., Fukushima, M.: On the rate of convergence of the Levenberg-Marguardt method. Comput., Suppl (Wien) 15, 227--238 (2001) · Zbl 1001.65047