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.     

Weaker conditions for the convergence of Newton’s method

Newton’s method is often used for solving nonlinear equations. In this paper, we show that Newton’s method converges under weaker convergence criteria than those given in earlier studies, such as Argyros (2004) [2, p. 387], Argyros and Hilout (2010)[11, p. 12], Argyros et al. (2011) [12, p. 26], Ortega and Rheinboldt (1970) [26, p. 421], Potra and Pták (1984) [36, p. 22]. These new results are illustrated by several numerical examples, for which the older convergence criteria do not hold but for which our weaker convergence criteria are satisfied.     

Convergence of a generalized Newton and an inexact generalized Newton algorithms for solving nonlinear equations with nondifferentiable terms

In this paper, we consider two versions of the Newton-type method for solving a nonlinear equations with nondifferentiable terms, which uses as iteration matrices, any matrix from B-differential of semismooth terms. Local and global convergence theorems for the generalized Newton and inexact generalized Newton method are proved. Linear convergence of the algorithms is obtained under very mild assumptions. The superlinear convergence holds under some conditions imposed on both terms of equation. Some numerical results indicate that both algorithms works quite well in practice.      

A globally convergent inexact Newton method with a new choice for the forcing term

In inexact Newton methods for solving nonlinear systems of equations, an approximation to the step s _k of the Newton’s system J(x _k )s=−F(x _k ) is found. This means that s _k must satisfy a condition like ‖F(x _k )+J(x _k )s _k ‖≤η _k ‖F(x _k )‖ for a forcing term η _k ∈[0,1). Possible choices for η _k have already been presented. In this work, a new choice for η _k is proposed. The method is globalized using a robust backtracking strategy proposed by Birgin et al. (Numerical Algorithms 32:249–260, 2003), and its convergence properties are proved. Several numerical experiments with boundary value problems are presented. The numerical performance of the proposed algorithm is analyzed by the performance profile tool proposed by Dolan and Moré (Mathematical Programming Series A 91:201–213, 2002). The results obtained show a competitive inexact Newton method for solving academic and applied problems in several areas. Supported by FAPESP, CNPq, PRONEX-Optimization.     

A smoothing inexact Newton method for nonlinear complementarity problems

In this article, we propose a new smoothing inexact Newton algorithm for solving nonlinear complementarity problems (NCP) base on the smoothed Fischer-Burmeister function. In each iteration, the corresponding linear system is solved only approximately. The global convergence and local superlinear convergence are established without strict complementarity assumption at the NCP solution. Preliminary numerical results indicate that the method is effective for large-scale NCP.     

A unifying theorem for Newton’s method on spaces with a convergence structure

We present a semilocal convergence theorem for Newton’s method (NM) on spaces with a convergence structure. Using our new idea of recurrent functions, we provide a tighter analysis, with weaker hypotheses than before and with the same computational cost as for Argyros (1996, 1997, 1997, 2007) [1], [2], [3] and [5], Meyer (1984, 1987, 1992) [13], [14] and [15]. Numerical examples are provided for solving equations in cases not covered before.     

A Newton inexact interior-point method for large scale nonlinear optimization problems

In this paper, we describe a variant of the Newton Interior-Point method in [8] for nonlinear programming problems. In this scheme, the perturbation parameter can be chosen within a range of, values and we can use an iterative method for approximately solving the reduced linear system arising at each step. We have devised the inner termination rule which guarantees the global convergence of this Newton Inexact Interior-Point method. We remark that the required assumptions are weaker than those stated in [8], as shown by some numerical examples. This research was supported by the Italian Ministry for Education, University and Research (MIUR), FIRB Project No. RBAU01JYPN.     

A nonsmooth inexact Newton method for the solution of large-scale nonlinear complementarity problems

A new algorithm for the solation of large-scale nonlinear complementarity problems is introduced. The algorithm is based on a nonsmooth equation reformulation of the complementarity problem and on an inexact Levenberg-Marquardt-type algorithm for its solution. Under mild assumptions, and requiring only the approximate solution of a linear system at each iteration, the algorithm is shown to be both globally and superlinearly convergent, even on degenerate problems. Numerical results for problems with up to 10 000 variables are presented. Partially supported by Agenzia Spaziale Italiana, Roma, Italy.     

A unified proof of Brooks’ theorem and Catlin’s theorem

Brooks’ theorem is a fundamental result in the theory of graph coloring. Catlin proved the following strengthening of Brooks’ theorem: Let d$d$ be an integer at least 3, and let G$G$ be a graph with maximum degree d$d$. If G$G$ does not contain _Kd+1$K_{d+1}$ as a subgraph, then G$G$ has a d$d$-coloring in which one color class has size α(G)$\alpha \left(G\right)$. Here α(G)$\alpha \left(G\right)$ denotes the independence number of G$G$. We give a unified proof of Brooks’ theorem and Catlin’s theorem.     

A convergence theorem for the Newton-like methods under some kind of weak Lipschitz conditions

We provide sufficient conditions for the convergence of the Newton-like methods in the assumption that the derivative satisfies some kind of weak Lipschitz conditions. Consequently, some important convergence theorems follow from our main result in this paper.     

Local convergence of Newton’s method under majorant condition

A local convergence analysis of Newton’s method for solving nonlinear equations, under a majorant condition, is presented in this paper. Without assuming convexity of the derivative of the majorant function, which relaxes the Lipschitz condition on the operator under consideration, convergence, the biggest range for uniqueness of the solution, the optimal convergence radius and results on the convergence rate are established. Besides, two special cases of the general theory are presented as applications.     

Landau’s theorem for polyharmonic mappings

In this paper, we first investigate coefficient estimates for bounded polyharmonic mappings in the unit disk D$\mathbb{D}$. Then, we obtain two versions of Landau’s theorem for polyharmonic mappings F$F$, and for the mappings of the type L(F)$L\left(F\right)$, where L$L$ is the differential operator of Abdulhadi, Abu Muhanna and Khuri. Examples and numerical estimates are given.     

Fractional Noether’s theorem in the Riesz-Caputo sense

We prove a Noether’s theorem for fractional variational problems with Riesz-Caputo derivatives. Both Lagrangian and Hamiltonian formulations are obtained. Illustrative examples in the fractional context of the calculus of variations and optimal control are given.     

Improving the rate of convergence of Newton methods on Banach spaces with a convergence structure and applications

In this note, we use inexact Newton-like methods to find solutions of nonlinear operator equations on Banach spaces with a convergence structure. Our technique involves the introduction of a generalized norm as an operator from a linear space into a partially ordered Banach space. In this way, the metric properties of the examined problem can be analyzed more precisely. Moreover, this approach allows us to derive from the same theorem, on the one hand, semilocal results of Kantorovich-type, and on the other hand, global results based on monotonicity considerations. By imposing very general Lipschitz-like conditions on the operators involved, on the one hand, we cover a wider range of problems, and on the other hand, by choosing our operators appropriately, we can find sharper error bounds on the distances involved than before. Furthermore, we show that special cases of our results reduce to the corresponding ones already in the literature. Finally, several examples are being provided where our results compare favorably with earlier ones.     

A wavelet adaptive Newton method for the solution of nonlinear equations

For the solution of nonlinear equations, we present an adaptive wavelet scheme, which couples an inexact Newton method and the idea of nonlinear wavelet approximation. In particular, we obtain a result of quadratic convergence.     

A generalization of Sperner’s theorem and an application to graph orientations

A generalization of Sperner’s theorem is established: For a multifamily M={Y₁,…,Y_p} of subsets of {1,…,n} in which the repetition of subsets is allowed, a sharp lower bound for the number φ(M) of ordered pairs (i,j) satisfying i≠j and Y_i⊆Y_j is determined. As an application, the minimum average distance of orientations of complete bipartite graphs is determined.     

On convergence of the modified Newton’s method under Hölder continuous Fréchet derivative

In this paper, the upper and lower estimates of the radius of the convergence ball of the modified Newton’s method in Banach space are provided under the hypotheses that the Fréchet derivative of the nonlinear operator are center Hölder continuous for the initial point and the solution of the operator. The error analysis is given which matches the convergence order of the modified Newton’s method. The uniqueness ball of solution is also established. Numerical examples for validating the results are also provided, including a two point boundary value problem.     

Inexact Newton-type methods

We introduce the new idea of recurrent functions to provide a semilocal convergence analysis for an inexact Newton-type method, using outer inverses. It turns out that our sufficient convergence conditions are weaker than in earlier studies in many interesting cases (Argyros, 2004 [5] and [6], Argyros, 2007 [7], Dennis, 1971 [14], Deuflhard and Heindl, 1979 [15], Gutiérrez, 1997 [16], Gutiérrez et al., 1995 [17], Häubler, 1986 [18], Huang, 1993 [19], Kantorovich and Akilov, 1982 [20], Nashed and Chen, 1993 [21], Potra, 1982 [22], Potra, 1985 [23]).     

Generalized equations and the generalized Newton method

We give some convergence results on the generalized Newton method (referred to by some authors as Newton's method) and the chord method when applied to generalized equations. The main results of the paper extend the classical Kantorovich results on Newton's method to (nonsmooth) generalized equations. Our results also extend earlier results on nonsmooth equations due to Eaves, Robinson, Josephy, Pang and Chan. We also propose inner-iterative schemes for the computation of the generalized Newton iterates. These schemes generalize popular iterative methods (Richardson's method, Jacobi's method and the Gauss-Seidel method) for the solution of linear equations and linear complementarity problems and are shown to be convergent under natural generalizations of classical convergence criteria. Our results are applicable to equations involving single-valued functions and also to a class of generalized equations which includes variational inequalities, nonlinear complementarity problems and some nonsmooth convex minimization problems.     

Monge’s method for intermediate integrals

In this paper we will present a framework of solving a second order differential equation using a solvable first order equation. Such a first order equation is called an intermediate integral. Illustrative examples are used to highlight the proposed algorithm.