Kantorovich-type semilocal convergence analysis for inexact Newton methods

We provide a new semilocal convergence analysis for generating an inexact Newton method converging to a solution of a nonlinear equation in a Banach space setting. Our analysis is based on our idea of recurrent functions. Our results are compared favorably to earlier ones by others and us (Argyros (2007, 2009) [5] and [6], Argyros and Hilout (2009) [7], Guo (2007) [15], Shen and Li (2008) [18], Li and Shen (2008) [19], Shen and Li (2009) [20]). Numerical examples are provided to show that our results apply, but not earlier ones [15], [18], [19] and [20].     

A new semi-local convergence theorem for the inexact Newton methods

We present a new semi-local convergence theorem for the inexact Newton methods in the assumption that the derivative satisfies some kind of weak Lipschitz conditions. As special cases of our main result we re-obtain some well-known convergence theorems for Newton methods.     

On the semilocal convergence of inexact Newton methods in Banach spaces

We provide two types of semilocal convergence theorems for approximating a solution of an equation in a Banach space setting using an inexact Newton method [I.K. Argyros, Relation between forcing sequences and inexact Newton iterates in Banach spaces, Computing 63 (2) (1999) 134–144; I.K. Argyros, A new convergence theorem for the inexact Newton method based on assumptions involving the second Fréchet-derivative, Comput. Appl. Math. 37 (7) (1999) 109–115; I.K. Argyros, Forcing sequences and inexact Newton iterates in Banach space, Appl. Math. Lett. 13 (1) (2000) 77–80; I.K. Argyros, Local convergence of inexact Newton-like iterative methods and applications, Comput. Math. Appl. 39 (2000) 69–75; I.K. Argyros, Computational Theory of Iterative Methods, in: C.K. Chui, L. Wuytack (Eds.), in: Studies in Computational Mathematics, vol. 15, Elsevier Publ. Co., New York, USA, 2007; X. Guo, On semilocal convergence of inexact Newton methods, J. Comput. Math. 25 (2) (2007) 231–242]. By using more precise majorizing sequences than before [X. Guo, On semilocal convergence of inexact Newton methods, J. Comput. Math. 25 (2) (2007) 231–242; Z.D. Huang, On the convergence of inexact Newton method, J. Zheijiang University, Nat. Sci. Ed. 30 (4) (2003) 393–396; L.V. Kantorovich, G.P. Akilov, Functional Analysis, Pergamon Press, Oxford, 1982; X.H. Wang, Convergence on the iteration of Halley family in weak condition, Chinese Sci. Bull. 42 (7) (1997) 552–555; T.J. Ypma, Local convergence of inexact Newton methods, SIAM J. Numer. Anal. 21 (3) (1984) 583–590], we provide (under the same computational cost) under the same or weaker hypotheses: finer error bounds on the distances involved; an at least as precise information on the location of the solution. Moreover if the splitting method is used, we show that a smaller number of inner/outer iterations can be obtained.     

On the convergence of inexact Newton methods for discrete-time algebraic Riccati equations

In this paper, we present a convergence analysis of the inexact Newton method for solving Discrete-time algebraic Riccati equations (DAREs) for large and sparse systems. The inexact Newton method requires, at each iteration, the solution of a symmetric Stein matrix equation. These linear matrix equations are solved approximatively by the alternating directions implicit (ADI) or Smith?s methods. We give some new matrix identities that will allow us to derive new theoretical convergence results for the obtained inexact Newton sequences. We show that under some necessary conditions the approximate solutions satisfy some desired properties such as the d-stability. The theoretical results developed in this paper are an extension to the discrete case of the analysis performed by Feitzinger et al. (2009) [8] for the continuous-time algebraic Riccati equations. In the last section, we give some numerical experiments.     

On convergence rates of inexact Newton regularizations

Summary. REGINN is an algorithm of inexact Newton type for the regularization of nonlinear ill-posed problems [Inverse Problems 15 (1999), pp. 309–327]. In the present article convergence is shown under weak smoothness assumptions (source conditions). Moreover, convergence rates are established. Some computational illustrations support the theoretical results. Received March 12, 1999 / Published online October 16, 2000     

The convergence analysis of inexact Gauss–Newton methods for nonlinear problems

In this paper, inexact Gauss–Newton methods for nonlinear least squares problems are studied. Under the hypothesis that derivative satisfies some kinds of weak Lipschitz conditions, the local convergence properties of inexact Gauss–Newton and inexact Gauss–Newton like methods for nonlinear problems are established with the modified relative residual control. The obtained results can provide an estimate of convergence ball for inexact Gauss–Newton methods.     

Convergence behaviour of inexact Newton methods

In this paper we investigate local convergence properties of inexact Newton and Newton-like methods for systems of nonlinear equations. Processes with modified relative residual control are considered, and new sufficient conditions for linear convergence in an arbitrary vector norm are provided. For a special case the results are affine invariant.

     

Towards a general convergence theory for inexact Newton regularizations

We develop a general convergence analysis for a class of inexact Newton-type regularizations for stably solving nonlinear ill-posed problems. Each of the methods under consideration consists of two components: the outer Newton iteration and an inner regularization scheme which, applied to the linearized system, provides the update. In this paper we give a novel and unified convergence analysis which is not confined to a specific inner regularization scheme but applies to a multitude of schemes including Landweber and steepest decent iterations, iterated Tikhonov method, and method of conjugate gradients.     

Convergence of inexact Newton methods for generalized equations

For solving the generalized equation $f(x)+F(x) \ni 0$ , where $f$ is a smooth function and $F$ is a set-valued mapping acting between Banach spaces, we study the inexact Newton method described by $$\begin{aligned} \left( f(x_k)+ D f(x_k)(x_{k+1}-x_k) + F(x_{k+1})\right) \cap R_k(x_k, x_{k+1}) \ne \emptyset , \end{aligned}$$ where $Df$ is the derivative of $f$ and the sequence of mappings $R_k$ represents the inexactness. We show how regularity properties of the mappings $f+F$ and $R_k$ are able to guarantee that every sequence generated by the method is convergent either q-linearly, q-superlinearly, or q-quadratically, according to the particular assumptions. We also show there are circumstances in which at least one convergence sequence is sure to be generated. As a byproduct, we obtain convergence results about inexact Newton methods for solving equations, variational inequalities and nonlinear programming problems.     

Globally convergent Jacobian smoothing inexact Newton methods for NCP

A new smoothing algorithm for the solution of nonlinear complementarity problems (NCP) is introduced in this paper. It is based on semismooth equation reformulation of NCP by Fischer–Burmeister function and its related smooth approximation. In each iteration the corresponding linear system is solved only approximately. Since inexact directions are not necessarily descent, a nonmonotone technique is used for globalization procedure. Numerical results are also presented. Research supported by Ministry of Science, Republic of Serbia, grant No. 144006.     

Local convergence theorems for Newton methods

Affine invariant sufficient conditions are given for two local convergence theorems involving inexact Newton-like methods. The first uses conditions on the first Fréchet-derivative whereas the second theorem employs hypotheses on themth (m ≥ 2 an integer). Radius of convergence as well as rate of convergence results are derived. Results involving superlinear convergence and known to be true for inexact Newton methods are extended here. Moreover, we show that under hypotheses on the mth Fréchet-derivative our radius of convergence can sometimes be larger than the corresponding one in [10]. This allows a wider choice for the initial guess. A numerical example is also provided to show that our radius of convergence is larger than the one in [10].     

Convergence behaviour of inexact Newton methods under weak Lipschitz condition

Under weak Lipschitz condition, local convergence properties of inexact Newton methods and Newton-like methods for systems of nonlinear equations are established in an arbitrary vector norm. Processes with modified relative residual control are considered; the results easily provide an estimate of convergence ball for inexact methods. For a special case, the results are affine invariant. Some applications are given.     

Rates of convergence for adaptive Newton methods

We consider Newton-type methods for constrained optimization problems in infinite-dimensional spaces, where at each iteration the first and second derivatives and the feasible set are approximated. The approximations can change at each iteration and conditions are given under which linear and superlinear rates of convergence of the iterates to the optimal point hold. Several applications are discussed.     

Smoothing inexact Newton methods for NCP with various nonmonotone techniques

Various iterative methods for solving nonlinear complementarity problems (NCP) are developed in recent years. In this paper we propose Jacobian smoothing inexact Newton methods for NCP with different nonmonotone strategies. The methods are based on semismooth equation reformulation of NCP by Fischer-Burmeister function. Nonmonotone line-search techniques are used for globalization procedure. Numerical performance of algorithms are compared. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Affine invariant local convergence theorems for inexact Newton-like methods

Affine invariant sufficient conditions are given for two local convergence theorems involving inexact Newton-like methods. The first uses conditions on the first Fréchet-derivative whereas the second theorem employs hypotheses on the second. Radius of convergence as well as rate of convergence results are derived. Results involving superlinear convergence and known to be true for inexact Newton methods are extended here. Moreover, we show that under hypotheses on the second Fréchet-derivative our radius of convergence is larger than the corresponding one in [10]. This allows a wider choice for the initial guess. A numerical example is also provided to show that our radius of convergence is larger than the one in [10].     

Concerning the semilocal convergence of Newton’s method and convex majorants

We approximate a locally unique solution of an equation on a Banach space setting using Newton’smethod.Motivated by the work by Ferreira and Svaiter [5] but using more precise majorization sequences, and under the same computational cost we provide: a larger convergence region; finer error bounds on the distances involved, and an at least as precise information on the location of the solution than in [5]. The results can also compare favorably to the corresponding ones given byWang in [10]. Finally we complete the study with two concrete applications.      

An improved inexact Newton method

For unconstrained optimization, an inexact Newton algorithm is proposed recently, in which the preconditioned conjugate gradient method is applied to solve the Newton equations. In this paper, we improve this algorithm by efficiently using automatic differentiation and establish a new inexact Newton algorithm. Based on the efficiency coefficient defined by Brent, a theoretical efficiency ratio of the new algorithm to the old algorithm is introduced. It has been shown that this ratio is greater than 1, which implies that the new algorithm is always more efficient than the old one. Furthermore, this improvement is significant at least for some cases. This theoretical conclusion is supported by numerical experiments.      

Local convergence analysis of inexact Newton-like methods under majorant condition

We present a local convergence analysis of inexact Newton-like methods for solving nonlinear equations under majorant conditions. This analysis provides an estimate of the convergence radius and a clear relationship between the majorant function, which relaxes the Lipschitz continuity of the derivative, and the nonlinear operator under consideration. It also allow us to obtain some important special cases.     

Global convergence for Newton methods in mathematical programming

In constrained optimization problems in mathematical programming, one wants to minimize a functionalf(x) over a given setC. If, at an approximate solutionx _n , one replacesf(x) by its Taylor series expansion through quadratic terms atx _n and denotes byx _n+1 the minimizing point for this overC, one has a direct analogue of Newton's method. The local convergence of this has been previously analyzed; here, we give global convergence results for this and the similar algorithm in which the constraint setC is also linearized at each step.This research was supported in part by the Office of Naval Research, Contract No. N00014-67-0126-0015, and was presented by invitation at the Fifth Gatlinburg Symposium on Numerical Algebra, Los Alamos, New Mexico, 1972.