A family of Newton methods for nonsmooth constrained systems with nonisolated solutions

We propose a new family of Newton-type methods for the solution of constrained systems of equations. Under suitable conditions, that do not include differentiability or local uniqueness of solutions, local, quadratic convergence to a solution of the system of equations can be established. We show that as particular instances of the method we obtain inexact versions of both a recently introduced LP-based Newton method and of a Levenberg-Marquardt algorithm for the solution of systems with nonisolated solutions, and improve on corresponding existing results.     

On the convergence of the Galerkin method for nonsmooth solutions of integral equations

Summary It is shown that the stability region of the Galerkin method includes solutions not lying in the conventional energy space. Optimal order error estimates for these nonsmooth solutions are derived. The new result is compared with the classical statement by means of the basic potential problem.     

Local behavior of an iterative framework for generalized equations with nonisolated solutions

 An iterative framework for solving generalized equations with nonisolated solutions is presented. For generalized equations with the structure , where is a multifunction and F is single-valued, the framework covers methods that, at each step, solve subproblems of the type . The multifunction approximates F around s. Besides a condition on the quality of this approximation, two other basic assumptions are employed to show Q-superlinear or Q-quadratic convergence of the iterates to a solution. A key assumption is the upper Lipschitz-continuity of the solution set map of the perturbed generalized equation . Moreover, the solvability of the subproblems is required. Conditions that ensure these assumptions are discussed in general and by means of several applications. They include monotone mixed complementarity problems, Karush-Kuhn-Tucker systems arising from nonlinear programs, and nonlinear equations. Particular results deal with error bounds and upper Lipschitz-continuity properties for these problems. Received: November 2001 / Accepted: November 2002 Published online: December 9, 2002 Key Words. generalized equation – nonisolated solutions – Newton's method – superlinear convergence – upper Lipschitz-continuity – mixed complementarity problem – error bounds Mathematics Subject Classification (1991): 90C30, 65K05, 90C31, 90C33     

仿射变换内点Levenberg-Marquardt法解KKT系统

提供了一类新的结合非单调内点回代线搜索技术的仿射变换Levenberg-Marquardt法解Karush-Kuhn-Tucker（KKT）系统. 基于由KKT系统转化得到的等价的部分变量具有非负约束的最小化问题,建立了Levenberg-Marquardt方程. 证明了算法不仅具有整体收敛性,而且在合理的假设条件下,算法具有超线性收敛速率. 数值结果验证了算法的实际有效性.     

Singular solutions for semi-linear parabolic equations on nonsmooth domains

We prove the existence of positive singular solutions for the semi-linear parabolic equation on Ω=D×]0,∞[, where p>1,D is a bounded NTA-domain in Rn, n?2, and μ is in a general class of signed Radon measures on D covering the elliptic Kato class of potentials adopted by Zhang and Zhao. A new proof of the result based on a simple fixed point theorem is also given.     

A parameterized Newton method and a quasi-Newton method for nonsmooth equations

This paper presents a parameterized Newton method using generalized Jacobians and a Broyden-like method for solving nonsmooth equations. The former ensures that the method is well-defined even when the generalized Jacobian is singular. The latter is constructed by using an approximation function which can be formed for nonsmooth equations arising from partial differential equations and nonlinear complementarity problems. The approximation function method generalizes the splitting function method for nonsmooth equations. Locally superlinear convergence results are proved for the two methods. Numerical examples are given to compare the two methods with some other methods.This work is supported by the Australian Research Council.     

On the Secant method for solving nonsmooth equations

In this study we are concerned with the problem of approximating a locally unique solution of an operator equation in Banach space using the Secant method. The differentiability of the operator involved is not assumed. Using a flexible point-based approximation, we provide a local as well as a semilocal convergence analysis for the Secant method. Our results are justified by numerical examples that cannot be handled with earlier works.     

求解非光滑方程组的三次正则化方法

考虑求解非光滑方程组的三次正则化方法及其收敛性分析.利用信赖域方法的技巧,保证该方法是全局收敛的.在子问题非精确求解和BD正则性条件成立的前提下,分析了非光滑三次正则化方法的局部收敛速度.最后,数值实验结果验证了该算法的有效性.     

Strong positivity of solutions to parabolic and elliptic equations on nonsmooth domains

Nonnegativity of weak solutions of parabolic and elliptic equations on nonsmooth domains is established. Strong positivity of weak solutions to elliptic equations is proved via a boundary weak Harnack inequality.     

Multimode systems of nonlinear equations: Derivation,integrability, and numerical solutions

We consider the propagation of electromagnetic pulses in isotropic media taking a third-order nonlinearity into account. We develop a method for transforming Maxwell’s equations based on a complete set of projection operators corresponding to wave-dispersion branches (in a waveguide or in matter) with the propagation direction taken into account. The most important result of applying the method is a system of equations describing the one-dimensional dynamics of pulses propagating in opposite directions without accounting for dispersion. We derive the corresponding self-action equations. We thus introduce dispersion in the media and show how the operators change. We obtain generalized Schäfer-Wayne short-pulse equations accounting for both propagation directions. In the three-dimensional problem, we focus on optic fibers with dispersive matter, deriving and numerically solving equations of the waveguide-mode interaction. We discuss the effects of the interaction of unidirectional pulses. For the coupled nonlinear Schrödinger equations, we discuss a concept of numerical integrability and apply the developed calculation schemes.     

Periodic solutions for planar autonomous systems with nonsmooth periodic perturbations

In this paper we consider a class of planar autonomous systems having an isolated limit cycle x₀ of smallest period T>0 such that the associated linearized system around it has only one characteristic multiplier with absolute value 1. We consider two functions, defined by means of the eigenfunctions of the adjoint of the linearized system, and we formulate conditions in terms of them in order to have the existence of two geometrically distinct families of T-periodic solutions of the autonomous system when it is perturbed by nonsmooth T-periodic nonlinear terms of small amplitude. We also show the convergence of these periodic solutions to x₀ as the perturbation disappears and we provide an estimation of the rate of convergence. The employed methods are mainly based on the theory of topological degree and its properties that allow less regularity on the data than that required by the approach, commonly employed in the existing literature on this subject, based on various versions of the implicit function theorem.     

Multiple solutions for nonlinear elliptic equations at resonance with a nonsmooth potential

In this paper, we study a nonlinear elliptic problem at resonance driven by the p-Laplacian and with a nonsmooth potential (hemivariational inequality). Our approach is variational and it is based on the nonsmooth critical point theory for locally Lipschitz functions due to Chang. We prove a theorem guaranteeing the existence of one solution which is smooth and strictly positive. Then by strengthening the assumptions, we establish a multiplicity result providing the existence of at least two distinct solutions. Our hypotheses permit resonance and asymmetric behavior at +∞ and −∞. As a byproduct of our analysis we obtain an nonlinear and nonsmooth generalization of a result of Brézis–Nirenberg about H₀¹ versus C₀¹ minimizers of a smooth functional.     

On a new exponential iterative method for solving nonsmooth equations

In this paper, we propose a new distinctive version of a generalized Newton method for solving nonsmooth equations. The iterative formula is not the classic Newton type, but an exponential one. Moreover, it uses matrices from B‐differential instead of generalized Jacobian. We prove local convergence of the method and we present some numerical examples.     

Numerical solutions of systems of nonlinear integro-differential equations by Homotopy-perturbation method

In this study, the numerical solutions of a system of two nonlinear integro-differential equations, which describes biological species living together, are derived employing the well-known Homotopy-perturbation method. The approximate solutions are in excellent agreement with those obtained by the Adomian decomposition method. Furthermore, we present an analytical approximate solution for a more general form of the system of nonlinear integro-differential equations. The numerical result indicates that the proposed method is straightforward to implement, efficient and accurate for solving nonlinear integro-differential equations.     

Nodal and multiple constant sign solutions for resonant -Laplacian equations with a nonsmooth potential

In this paper we study a nonlinear Dirichlet elliptic differential equation driven by the p-Laplacian and with a nonsmooth potential (hemivariational inequality). Using a variational approach combined with suitable truncation techniques and the method of upper–lower solutions, we prove the existence of five nontrivial smooth solutions, two positive, two negative and the fifth nodal. Our hypotheses on the nonsmooth potential allow resonance at infinity with respect to the principal eigenvalue λ₁>0 of .     

An aggregate subgradient method for nonsmooth and nonconvex minimization

This paper presents a readily implementable algorithm for minimizing a locally Lipschitz continuous function that is not necessarily convex or differentiable. This extension of the aggregate subgradient method differs from one developed by the author in the treatment of nonconvexity. Subgradient aggregation allows the user to control the number of constraints in search direction finding subproblems and, thus, trade-off subproblem solution effort for rate of convergence. All accumulation points of the algorithm are stationary. Moreover, the algorithm converges when the objective function happens to be convex.