On the Convergence Analysis of a Two-Step Modification of the Gauss-Newton Method

We investigate the convergence of a two-step modification of the Gauss-Newton method applying the generalized Lipschitz condition for the first and second order derivatives. The convergence order as well as the convergence radius of the method are studied and the uniqueness ball of solution of the nonlinear least squares problem is examined. Finally, we carry out numerical experiments on a set of well-known test problems. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On an one-step modification of Gauss-Newton method under generalized Lipschitz conditions for solving the nonlinear least squares problem

We investigate convergence of the one-step modification of Gauss-Newton method using the divided differences and the weak generalized Lipschitz condition for the divided differences. Convergence order of the method was examined and the uniqueness ball for the solution of the nonlinear least squares problem was proved. Numerical experiments were also provided. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Local convergence analysis of the Gauss-Newton method under a majorant condition

The Gauss-Newton method for solving nonlinear least squares problems is studied in this paper. Under the hypothesis that the derivative of the function associated with the least square problem satisfies a majorant condition, a local convergence analysis is presented. This analysis allows us to obtain the optimal convergence radius and the biggest range for the uniqueness of stationary point, and to unify two previous and unrelated results.     

Convergence of the Gauss-Newton method for convex composite optimization problems under majorant condition on Riemannian manifolds

In this paper, we consider convex composite optimization problems on Riemannian manifolds, and discuss the semi-local convergence of the Gauss-Newton method with quasi-regular initial point and under the majorant condition. As special cases, we also discuss the convergence of the sequence generated by the Gauss-Newton method under Lipschitz-type condition, or under γ-condition.     

Existence and Uniqueness of Smooth Solution for a Four-waves Coupled System

In this paper, we consider a four-waves coupled system which describes the interaction between particles. Based on the uniform bound and strong convergence property in lower order norm, local existence and uniqueness of smooth solution is established by a limiting argument. Moreover, we show the solution exists globally in two dimensional case under certain condition on the size for $L^2$ norm of the initial data.     

Accelerated Gauss-Newton algorithms for nonlinear least squares problems

Recent theoretical and practical investigations have shown that the Gauss-Newton algorithm is the method of choice for the numerical solution of nonlinear least squares parameter estimation problems. It is shown that when line searches are included, the Gauss-Newton algorithm behaves asymptotically like steepest descent, for a special choice of parameterization. Based on this a conjugate gradient acceleration is developed. It converges fast also for those large residual problems, where the original Gauss-Newton algorithm has a slow rate of convergence. Several numerical test examples are reported, verifying the applicability of the theory.     

非线性最小二乘问题收敛性的证明

主要讨论了无约束最优化中非线性最小二乘问题的收敛性.侧重于收敛的速率和整体、局部分析.改变了Gauss—Newton方法收敛性定理的条件,分两种情况证明了:(1)目标函数的海赛矩阵正定(函数严格凸)时为强整体二阶收敛;(2)目标函数不保证严格凸性,但海赛矩阵的逆存在时为局部收敛,敛速仍为二阶,同时给出了J(X)~(-1)和Q(X)~(-1)之间存在、有界性的等价条件.     

Inexact Gauss-Newton like methods for injective-overdetermined systems of equations under a majorant condition

In this paper, inexact Gauss-Newton like methods for solving injective-overdetermined systems of equations are studied. We use a majorant condition, defined by a function whose derivative is not necessarily convex, to extend and improve several existing results on the local convergence of the Gauss-Newton methods. In particular, this analysis guarantees the convergence of the methods for two important new cases.     

Lösungen der poissongleichung und harmonische vektorfelder in unbeschränkten gebieten

In the first part of this paper we consider generalised solutions of the Poisson equation Δ U = F in open subsets of R _n(n ? 3) with Dirichlet or Neumann boundary data. We prove existence and uniqueness theorems, not only for the corresponding interior and exterior problems, but also for domains with boundaries extending to infinity. In the second part we discuss generalised harmonic fields in open subsets of R ₃ with vanishing Dirichlet or Neumann boundary condition.     

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

In this paper, we present a local convergence analysis of inexact Gauss-Newton like methods for solving nonlinear least squares problems. Under the hypothesis that the derivative of the function associated with the least squares problem satisfies a majorant condition, we obtain that the method is well-defined and converges. Our analysis provides a clear relationship between the majorant function and the function associated with the least squares problem. It also allows us to obtain an estimate of convergence ball for inexact Gauss-Newton like methods and some important, special cases.     

Positive weak solutions of semilinear second order elliptic inequalities via variational inequalities

Existence, uniqueness and convergence of approximants of positive weak solutions for semilinear second order elliptic inequalities are obtained. The nonlinearities involved in these inequalities satisfy suitable upper or lower bound conditions or monotonicity conditions. The lower bound conditions are allowed to contain the critical Sobolev exponents. The methodology is to establish variational inequality principles for demicontinuous pseudo-contractive maps in Hilbert spaces by considering convergence of approximants and apply them to the corresponding variational inequalities arising from the semilinear second order elliptic inequalities. Examples on the existence, uniqueness and convergence of approximants of positive weak solutions of the semilinear second order elliptic inequalities are given.     

A globalization scheme for the generalized Gauss-Newton method

Summary For solving an equality constrained nonlinear least squares problem, a globalization scheme for the generalized Gauss-Newton method via damping is proposed. The stepsize strategy is based on a special exact penalty function. Under natural conditions the global convergence of the algorithm is proved. Moreover, if the algorithm converges to a solution having a sufficiently small residual, the algorithm is shown to change automatically into the undamped generalized Gauss-Newton method with a fast linear rate of convergence. The behaviour of the method is demonstrated on hand of some examples taken from the literature.     

The numerical solution of the inverse Stefan problem

Summary The inverse Stefan problem can be understood as a problem of nonlinear approximation theory which we solved numerically by a generalized Gauss-Newton method introduced by Osborne and Watson [19]. Under some assumptions on the parameter space we prove its quadratic convergence and demonstrate its high efficiency by three numerical examples.     

Convergence analysis of the general Gauss-Newton algorithm

Summary The convergence of the Gauss-Newton algorithm for solving discrete nonlinear approximation problems is analyzed for general norms and families of functions. Aquantitative global convergence theorem and several theorems on the rate of local convergence are derived. A general stepsize control procedure and two regularization principles are incorporated. Examples indicate the limits of the convergence theorems.     

On convergence of Lavrentiev regularization for semilinear parabolic optimal control problems with pointwise control and state constraints

We consider Lavrentiev regularization for a class of semilinear parabolic optimal control problems with control constraints and pointwise state constraints and review convergence results for local solutions under Slater type assumptions as well as quadratic growth conditions. Moreover, we state a local uniqueness result for local optima under the assumptions of strict separability of the active sets as well as a second order sufficient condition for the regularized solution. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Existence and uniqueness results for a class of nonlinear differential systems with second-order differences

In a real Hilbert space we investigate the existence, uniqueness and asymptotic properties of the strong and weak solutions to an infinite nonlinear differential system with second–order differences, subject to an extreme condition and initial data. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Properties of equation reformulation of the Karush–Kuhn–Tucker condition for nonlinear second order cone optimization problems

We give an equation reformulation of the Karush–Kuhn–Tucker (KKT) condition for the second order cone optimization problem. The equation is strongly semismooth and its Clarke subdifferential at the KKT point is proved to be nonsingular under the constraint nondegeneracy condition and a strong second order sufficient optimality condition. This property is used in an implicit function theorem of semismooth functions to analyze the convergence properties of a local sequential quadratic programming type (for short, SQP-type) method by Kato and Fukushima (Optim Lett 1:129–144, 2007). Moreover, we prove that, a local solution x* to the second order cone optimization problem is a strict minimizer of the Han penalty merit function when the constraint nondegeneracy condition and the strong second order optimality condition are satisfied at x*.     

Semilocal convergence of multi-point improved super-Halley-type methods without the second derivative under generalized weak condition

In this paper, we consider the semilocal convergence of multi-point improved super-Halley-type methods in Banach space. Different from the results of super-Halley method studied in reference Gutiérrez, J.M. and Hernández, M.A. (Comput. Math. Appl. 36,1–8, 1998) these methods do not require second derivative of an operator, the R-order is improved and the convergence condition is also relaxed. We prove a convergence theorem to show existence and uniqueness of the solution.     

非Lipschitz条件下由Lévy过程驱动的倒向随机微分方程解的存在唯一及其稳定性

本文研究了由满足某种矩条件下Lévy过程相应的Teugel鞅及与之独立的布朗运动驱动的倒向随机微分方程,给出了飘逸系数满足非Lipschitz条件下解的存在唯一及稳定性结论.解的存在性是通过Picard迭代法给出的.解的L2收敛性是在飘逸系数弱于L2收敛意义下所得到的.     

On the theorem of T. Yamada and S. Watanabe

The well-known theorem of T. Yamada and S. Watanabe asserts that (weak) existence and pathwise uniqueness of the solution of a stochastic equation implies the existence of a strong solution. This is the most powerful tool for proving that a stochastic equation possesses a strong solution. However, pathwise uniqueness is far from being a necessary condition for this. Even if the solution is not unique in law it is also of interest to ask for strong solutions. In the present note, we will discuss in more detail the connection between pathwise uniqueness and the existence of a strong solution. We will state a condition which is not only sufficient but also necessary for the existence of a strong solution.