Convergence criterion and convergence ball of the Newton-type method in Banach space

The paper is concerned with the convergence problem of third-order Newton-type methods for finding zeros of nonlinear operations in Banach spaces. Under the hypothesis that the derivative of f satisfies the weak Lipschitz condition with L-average, the convergence criterion and convergence ball are given. Furthermore, some corollaries are obtained by applying the main results to some special functions L.     

Convergence analysis of a modified BFGS method on convex minimizations

A modified BFGS method is proposed for unconstrained optimization. The global convergence and the superlinear convergence of the convex functions are established under suitable assumptions. Numerical results show that this method is interesting.     

Convergence ball and error analysis of Ostrowski-Traub’s method

Under the hypotheses that the second-order and third-order derivatives of a function are bounded, an estimate of the radius of the convergence ball of Ostrowski-Traub’s method is obtained. An error analysis is given which matches the convergence order of the method. Finally, two examples are provided to show applications of our theorem.     

Convergence of a simple subgradient level method

Received December 3, 1996 / Revised version received January 29, 1998 Published online October 21, 1998     

Structure of the Hessian matrix and an economical implementation of Newton’s method in the problem of canonical approximation of tensors

A tensor given by its canonical decomposition is approximated by another tensor (again, in the canonical decomposition) of fixed lower rank. For this problem, the structure of the Hessian matrix of the objective function is analyzed. It is shown that all the auxiliary matrices needed for constructing the quadratic model can be calculated so that the computational effort is a quadratic function of the tensor dimensionality (rather than a cubic function as in earlier publications). An economical version of the trust region Newton method is proposed in which the structure of the Hessian matrix is efficiently used for multiplying this matrix by vectors and for scaling the trust region. At each step, the subproblem of minimizing the quadratic model in the trust region is solved using the preconditioned conjugate gradient method, which is terminated if a negative curvature direction is detected for the Hessian matrix.     

An improved local convergence analysis for Newton–Steffensen-type method

We provide a local convergence analysis for Newton–Steffensen-type algorithm for solving nonsmooth perturbed variational inclusions in Banach spaces. Under new center–conditions and the Aubin continuity property, we obtain the linear local convergence of Newton–Steffensen method. Our results compare favorably with related obtained in (Argyros and Hilout, 2007 submitted; Hilout in J. Math. Anal. Appl. 339:753–761, 2008).     

Convergence of Algorithms in Optimization and Solutions of Nonlinear Equations

It is desirable that an algorithm in unconstrained optimization converges when the guessed initial position is anywhere in a large region containing a minimum point. Furthermore, it is useful to have a measure of the rate of convergence which can easily be computed at every point along a trajectory to a minimum point. The Lyapunov function method provides a powerful tool to study convergence of iterative equations for computing a minimum point of a nonlinear unconstrained function or a solution of a system of nonlinear equations. It is surprising that this popular and powerful tool in the study of dynamical systems is not used directly to analyze the convergence properties of algorithms in optimization. We describe the Lyapunov function method and demonstrate how it can be used to study convergence of algorithms in optimization and in solutions of nonlinear equations. We develop an index which can measure the rate of convergence at all points along a trajectory to a minimum point and not just at points in a small neighborhood of a minimum point. Furthermore this index can be computed when the calculations are being carried out.     

A new analysis of electrostatic free energy minimization and Poisson–Boltzmann equation for protein in ionic solvent

In this paper, a novel solution decomposition of the Poisson dielectric model is proposed to modify a traditional electrostatic free energy minimization problem into one that is well defined for the case of protein in ionic solvent. The target function of this modified problem is shown to be strictly convex, weak sequentially lower semicontinuous, and twice continuously Fréchet differentiable. Its first and second Gâteaux derivatives are then found. Moreover, it is proved that this modified electrostatic free energy minimization problem has a unique solution, and its solution existence and uniqueness is equivalent to that of the Poisson–Boltzmann equation, a widely-used implicit solvent model for computing the electrostatics of biomolecules.     

Convergence Properties of the Regularized Newton Method for the Unconstrained Nonconvex Optimization

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.     

Convergence analysis of finite element approximations of the Joule heating problem in three spatial dimensions

In this paper we present a finite element discretization of the Joule-heating problem. We prove existence of solution to the discrete formulation and strong convergence of the finite element solution to the weak solution, up to a sub-sequence. We also present numerical examples in three spatial dimensions. The first example demonstrates the convergence of the method in the second example we consider an engineering application.     

A modified BFGS method and its superlinear convergence in nonconvex minimization with general line search rule

In this paper, a modification of the BFGS algorithm for unconstrained nonconvex optimization is proposed. The idea of the algorithm is to modify the approximate Hessian matrix for obtaining the descent direction and guaranteeing the efficacious of the new quasi-Newton iteration equation B _k+1 s _k =y _k ^* , where y _k ^* is the sum of y _k and t _k ‖g(x _k )‖s _k . The global convergence property of the algorithm associated with the general line search rule is prove.     

Convergence of Newton’s Method for Sections on Riemannian Manifolds

The present paper is concerned with the convergence problems of Newton’s method and the uniqueness problems of singular points for sections on Riemannian manifolds. Suppose that the covariant derivative of the sections satisfies the generalized Lipschitz condition. The convergence balls of Newton’s method and the uniqueness balls of singular points are estimated. Some applications to special cases, which include the Kantorovich’s condition and the γ-condition, as well as the Smale’s γ-theory for sections on Riemannian manifolds, are given. In particular, the estimates here are completely independent of the sectional curvature of the underlying Riemannian manifold and improve significantly the corresponding ones due to Dedieu, Priouret and Malajovich (IMA J. Numer. Anal. 23:395–419, 2003), as well as the ones in Li and Wang (Sci. China Ser. A. 48(11):1465–1478, 2005).     

A note on inexact gradient and Hessian conditions for cubic regularized Newton’s method

The inexact cubic-regularized Newton’s method (CR) proposed by Cartis, Gould and Toint achieves the same convergence rate as exact CR proposed by Nesterov and Polyak, but the inexact condition is not implementable due to its dependence on a future variable. This note establishes the same convergence rate under a similar but implementable inexact condition, which depends on only current variables. Our proof bounds the function-value decrease over total iterations rather than each iteration in the previous studies.     

On Tightness and Weak Convergence in the Approximation of the Occupation Measure of Fractional Brownian Motion

In this paper we consider approximations of the occupation measure of the Fractional Brownian motion by means of some functionals defined on regularizations of the paths. In a previous article Berzin and León proved a cylindrical convergence to a Wiener process of conveniently rescaled functionals. Here we show the tightness of the approximation in the space of continuous functions endowed with the topology of uniform convergence on compact sets. This allows us to simplify the identification of the limit.     

Convergence of Newton‘’s Method and Uniqueness of the Solution of Equations in Banach SpacesⅡ

Some results on convergence of Newton‘s method in Banach spaces are established under the assumption that the derivative of the opderators satisfies the radius or center Lipschitz condition with a weak L average.     

On convergence of homotopy analysis method and its modification for fractional modified KdV equations

In this paper, the homotopy analysis method (HAM) and its modification (MHAM) are applied to solve the nonlinear time- and space-fractional modified Korteweg-de Vries (fmKdV). The fractional derivatives are described by Caputo’s sense. Approximate and exact analytical solutions of the fmKdV are obtained. The MHAM in particular overcomes the computing difficulty encountered in HAM. Convergence theorems for both the homogeneous and non-homogeneous cases are given. The results of applying this procedure to the studied cases show the high accuracy and efficiency of the approach.     

Analysis of a smoothing Newton method for second-order cone complementarity problem

In this paper, we consider the second-order cone complementarity problem with P ₀-property. By introducing a smoothing parameter into the Fischer-Burmeister function, we present a smoothing Newton method for the second-order cone complementarity problem. The proposed algorithm solves only a linear system of equations and performs only one line search at each iteration. At the same time, the algorithm does not have restrictions on its starting point and has global convergence. Under the assumption of nonsingularity, we establish the locally quadratic convergence of the algorithm without strict complementarity condition. Preliminary numerical results show that the algorithm is promising.     

On the Convergence of Newton’s Method for Polynomial Equations and Applications in Radiative Transfer

 Newton’s method is used to approximate a locally unique zero of a polynomial operator F of degree in Banach space. So far, convergence conditions have been found for Newton’s method based on the Newton-Kantorovich hypothesis that uses Lipschitz-type conditions and information only on the first Fréchet-derivative of F. Here we provide a new semilocal convergence theorem for Newton’s method that uses information on all Fréchet-derivatives of F except the first. This way, we obtain sufficient convergence conditions different from the Newton-Kantorovich hypothesis. Our results are extended to include the case when F is a nonlinear operator whose kth Fréchet-derivative satisfies a H?lder continuity condition. An example is provided to show that our conditions hold where all previous ones fail. Moreover, some applications of our results to the solution of polynomial systems and differential equations are suggested. Furthermore, our results apply to solve a nonlinear integral equation appearing in radiative transfer in connection with the problem of determination of the angular distribution of the radiant-flux emerging from a plane radiation field. Received 9 December 1997 in revised form 30 March 1998