Local Convergence Analysis Research Articles

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.

On the local convergence of inexact Newton-type methods under residual control-type conditions

A local convergence analysis of inexact Newton-type methods using a new type of residual control was recently presented by C. Li and W. Shen. Here, we introduce the center-Hölder condition on the operator involved, and use it in combination with the Hölder condition to provide a new local convergence analysis with the following advantages: larger radius of convergence, and tighter error bounds on the distances involved. These results are obtained under the same hypotheses and computational cost. Numerical examples further validating the theoretical results are also provided in this study.

Local convergence of the proximal point method for a special class of nonconvex functions on Hadamard manifolds

Local convergence analysis of the proximal point method for a special class of nonconvex functions on Hadamard manifold is presented in this paper. The well definedness of the sequence generated by the proximal point method is guaranteed. Moreover, it is proved that each cluster point of this sequence satisfies the necessary optimality conditions and, under additional assumptions, its convergence for a minimizer is obtained.

Further Results for Perron–Frobenius Theorem for Nonnegative Tensors

For a nonnegative irreducible tensor, we give distribution properties of its eigenvalues. In particular, the spectral radius of a nonnegative irreducible tensor with positive trace is proved to be the unique eigenvalue on the spectral circle. Unlike the matrix setting, we give an example to present that this type of tensor is not always primitive. Thus, for a nonnegative irreducible tensor, the primitivity is a sufficient condition only for the spectral radius to be the unique eigenvalue on the spectral circle. Also, the stochastic tensor is defined, and we show that every nonnegative irreducible tensor with unit spectral radius is diagonally similar to a certain irreducible stochastic tensor. Based on this result, the minimax theorem for tensors is proved by using an alternative approach. Further, with the help of the minimax theorem, we illustrate that the problem of finding the spectral radius (largest singular value) of a nonnegative irreducible square (rectangular) tensor can be converted into a convex optimization problem. Additionally, we give an equivalent condition of irreducible nonnegative tensors. By this condition, one can easily determine whether or not a nonnegative tensor is irreducible.

LOCAL CONVERGENCE ANALYSIS OF INEXACT NEWTON-LIKE METHODS

We provide a local convergence analysis of inexact Newton–like methods in a Banach space setting under flexible majorant conditions. By introducing center–Lipschitz–type condition, we provide (under the same computational cost) a convergence analysis with the following advantages over earlier work [9]: finer error bounds on the distances involved, and a larger radius of convergence. Special cases and applications are also provided in this study.

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.

On an improved local convergence analysis for the Secant method

We provide a local convergence analysis for the Secant method in a Banach space setting under Holder continuous conditions. Using more precise estimates, and under the same computational cost, we enlarge the radius of convergence obtained in Ren and Wu (J Comput Appl Math 194:284–293, 2006).

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).

A TRAJECTORY-FOLLOWING METHOD FOR SOLVING THE STEADY STATE OF CHEMICAL REACTION RATE EQUATIONS

This article gives a trajectory-following method for the steady state of chemical reaction rate equations. In order to avoid wasting unnecessary computing time during the steady-state phase and get roughly accurate solutions during the transient-state phase, this method is realized via adopting the semi-implicit Euler formula as the stepping direction and adaptively adjusting the time-step size by an analogous trust-region technique. Under some standard assumptions, its global convergence analysis and local superlinear convergence analysis are also given. Finally, some numerical experiments of this method, in comparison with the traditional optimization methods and ordinary differential equation methods, are reported. The numerical results show that this trajectory-following method is a promising solution for this class of problems.

Generalized systems for relaxed cocoercive variational inequalities and projection methods in Hilbert spaces

In this paper, we introduce a new algorithm for a generalized system for a relaxed cocoercive nonlinear inequality and an asymptotically nonexpansive mapping in Hilbert spaces by the convergence of projection methods. Our results include the previous results as special cases extend and improve the main results of [R.U. Verma, General convergence analysis for two-step projection methods and application to variational problems. Appl. Math. Lett. 18 (11) (2005), 1286-1292], [R.U. Verma, Generalized system for relaxed cocoercive variational inequalities and its projection methods, J. Optim. Theory Appl. 121 (1) (2004), 203-210], [R.U. Verma, Generalized class of partial relaxed monotonicity and its connections, Adv. Nonlinear Var. Inequal. 7 (2) (2004), 155-164], [N.H. Xiu, J.Z. Zhang, Local convergence analysis of projection type algorithms: Unified approach, J. Optim. Theory Appl. 115 (2002) 211-230], [N.H. Nie, Z. Liu, K.H. Kim, S.M. Kang, A system of nonlinear variational inequalities involving strong monotone and pseudocontractive mappings, Adv. Nonlinear Var. Inequal. 6 (2) (2003), 91-99], [S.S. Chang, H.W. Joseph Lee, C.K. Chan, Generalized system for relaxed cocoercive variational inequalities in Hilbert spaces, Appl. Math. Lett. 20 (3) (2007), 329-334] and many others. Mathematics subject classification (2000): 47J05; 47J25.

LOCAL CONVERGENCE ANALYSIS FOR A CERTAIN CLASS OF INEXACT METHODS

We provide a local convergence analysis for a certain class inexact methods in a Banach space setting, in order to approximate a solution of a nonlinear equation [6]. The assumptions involve center-Lipschitz-type and radius-Lipschitz-type conditions [15], [8], [5]. Our results have the following advantages (under the same computational cost): larger radii, and finer error bounds on the distances involved than in [8], [15] in many interesting cases. Numerical examples further validating the theoretical results are also provided in this study.

ON THE LOCAL CONVERGENCE OF A TWO-STEP STEFFENSEN-TYPE METHOD FOR SOLVING GENERALIZED EQUATIONS

We use a two-step Steffensen-type method [1], [2], [4], [6], [13]-[16] to solve a generalized equation in a Banach space setting under Holder-type conditions introduced by us in [2], [6] for nonlinear equations. Using some ideas given in [4], [6] for nonlinear equations, we provide a local convergence analysis with the following advantages over related [13]-[16]: finer error bounds on the distances involved, and a larger radius of convergence. An application is also provided.

An improved local convergence analysis for a two-step Steffensen-type method

We study a class of Steffensen-type algorithm for solving nonsmooth variational inclusions in Banach spaces. We provide a local convergence analysis under ω-conditioned divided difference, and the Aubin continuity property. This work on the one hand extends the results on local convergence of Steffensen’s method related to the resolution of nonlinear equations (see Amat and Busquier in Comput. Math. Appl. 49:13–22, 2005; J. Math. Anal. Appl. 324:1084–1092, 2006; Argyros in Southwest J. Pure Appl. Math. 1:23–29, 1997; Nonlinear Anal. 62:179–194, 2005; J. Math. Anal. Appl. 322:146–157, 2006; Rev. Colomb. Math. 40:65–73, 2006; Computational Theory of Iterative Methods, 2007). On the other hand our approach improves the ratio of convergence and enlarges the convergence ball under weaker hypotheses than one given in Hilout (Commun. Appl. Nonlinear Anal. 14:27–34, 2007).

Newton’s method for approximating zeros of vector fields on Riemannian manifolds

We provide a local convergence analysis of Newton’s method for approximating zeros of vector fields on Riemannian manifolds. Using a combination of the radius and center-Lipschitz-type conditions on the covariant derivatives of the vector fields involved, we provide (under the same computational cost) a convergence analysis with following advantages over the corresponding one in (Li and Wang in Sci. China Ser. 48:1465–1478, 2005): at least as: large radius of convergence; small ratio of convergence. As applications, we extend the applicability of Kantorovich’s type result (Argyros in J. Comput. Appl. Math. 169:315–332, 2004; Argyros in J. Math. Anal. Appl. 298:374–397, 2004; Argyros in J. Appl. Math. Comput. 25: 35–47, 2007; Argyros in J. Concr. Appl. Anal. 6(1):21–32, 2008; Argyros in Convergence and Applications of Newton-Type Iterations, 2008; Ferreira and Svaiter in J. Complex. 18:304–329, 2002; Kantorovich and Akilov in Functional Analysis in Normed Spaces, 1982), and the Smale gamma theory (Argyros in Convergence and Applications of Newton-Type Iterations, 2008; Smith in Fields Inst. Commun. vol. 3, pp. 113–146, 1994; Wang in IMA J. Numer. Anal. 20:123–134, 2000).

On the radius of convergence of Newton’s method under average mild differentiability conditions

We provide a local convergence analysis for Newton’s method under mild differentiability conditions on the operator involved in a Banach space setting. In particular we show that under the same hypotheses and computational cost but using more precise estimates we can provide a larger convergence radius and finer error bounds on the distances involved than before (Huang in Comput. Math. Math. 42:247–251, 2004; Rheinboldt in Banach Ctr. Publ. 3:129–142, 1977; Wang in IMA J. Numer. Anal. 20:123–134, 2000). Some numerical examples are used to further justify the usage of our results over the earlier ones mentioned above.

Local convergence of Newton's method in Banach space from the viewpoint of the majorant principle

A local convergence analysis of Newton's method for solving nonlinear equations, based on Kantorovich's majorant principle, is presented in this paper. This analysis provides a clear relationship between the majorant function, which relaxes the Lipschitz continuity of the derivative, and the nonlinear operator under consideration. It also allows us to obtain the optimal convergence radius, the biggest range for the uniqueness of the solution, and to unify some previous and unrelated results.

Global convergence of quasi-Newton methods based on adjoint Broyden updates

In this paper we introduce a quasi-Newton method for the solution of systems of non-linear equations based on the nested application of adjoint Broyden updates. In combination with a suitable line search this method yields convergence of the iteration under the same requirements on F as Newton's method. The successive use of adjoint Broyden updates yields even local r-linear convergence of the iteration and provides the requirements for the local convergence analysis that gives q-superlinear convergence of the iteration.

Local Convergence Analysis of FastICA and Related Algorithms

The FastICA algorithm is one of the most prominent methods to solve the problem of linear independent component analysis (ICA). Although there have been several attempts to prove local convergence properties of FastICA, rigorous analysis is still missing in the community. The major difficulty of analysis is because of the well-known sign-flipping phenomenon of FastICA, which causes the discontinuity of the corresponding FastICA map on the unit sphere. In this paper, by using the concept of principal fiber bundles, FastICA is proven to be locally quadratically convergent to a correct separation. Higher order local convergence properties of FastICA are also investigated in the framework of a scalar shift strategy. Moreover, as a parallelized version of FastICA, the so-called QR FastICA algorithm, which employs the QR decomposition (Gram-Schmidt orthonormalization process) instead of the polar decomposition, is shown to share similar local convergence properties with the original FastICA.

A globally convergent BFGS method for nonlinear monotone equations without any merit functions

r. Since 1965, there has been significant progress in the theoretical study on quasi-Newton methods for solving nonlinear equations, especially in the local convergence analysis. However, the study on global convergence of quasi-Newton methods is relatively fewer, especially for the BFGS method. To ensure global convergence, some merit function such as the squared norm merit function is typically used. In this paper, we propose an algorithm for solving nonlinear monotone equations, which combines the BFGS method and the hyperplane projection method. We also prove that the proposed BFGS method converges globally if the equation is monotone and Lipschitz continuous without differentiability requirement on the equation, which makes it possible to solve some nonsmooth equations. An attractive property of the proposed method is that its global convergence is independent of any merit function.We also report some numerical results to show efficiency of the proposed method.

A Nonlinear Iterative Reconstruction and Analysis Approach to Shape-Based Approximate Electromagnetic Tomography

A nonlinear Helmholtz-equation-modeled electromagnetic tomographic reconstruction problem is solved for the object boundary and inhomogeneity parameters in a damped Tikhonov-regularized Gauss-Newton (DTRGN) solution framework. In this paper, the object is represented in a suitable global basis, whereas the boundary is expressed as the zero level set of a signed-distance function. For an explicit parameterized boundary-representation-based reconstruction scheme, analytical Jacobian and Hessian calculations are made to express the changes in scattered field values w.r.t. changes in the inhomogeneity parameters and the control points in a spline representation of the object boundary, via the use of a level-set representation of the object. Even though, in this paper, a homogeneous dielectric is considered and a spline representation has been used to represent the boundary, the formulation can be used for a general global basis representation of the inhomogeneity as well as arbitrary parameterizations of the boundary, and is generalizable to three dimensions. Reconstruction results are presented for test cases of landminelike dielectric objects embedded in the ground under noisy data conditions. To confirm convergence and, at times, to know which of the obtained iterates are closer to the actual unknown solution, using a perturbation theory framework, a local (Hessian-based) convergence analysis is applied to the DTRGN scheme for the reconstruction, yielding estimates of convergence rates in the residual and parameter spaces.