A Trust Region Method for Optimization Problem with Singular Solutions

In this paper, we propose a trust region method for minimizing a function whose Hessian matrix at the solutions may be singular. The global convergence of the method is obtained under mild conditions. Moreover, we show that if the objective function is LC ² function, the method possesses local superlinear convergence under the local error bound condition without the requirement of isolated nonsingular solution. This is the first regularized Newton method with trust region technique which possesses local superlinear (quadratic) convergence without the assumption that the Hessian of the objective function at the solution is nonsingular. Preliminary numerical experiments show the efficiency of the method. This work is partly supported by the National Natural Science Foundation of China (Grant Nos. 70302003, 10571106, 60503004, 70671100) and Science Foundation of Beijing Jiaotong University (2007RC014).     

等式约束非凸优化问题的修正牛顿算法

本文设计了一个新的求解等式约束非凸优化问题的修正牛顿算法.利用修正的拉格朗日函数,通过求解线性方程组获得搜索方向,利用价值函数的线性近似模型确定步长.在没有非奇异性假设的条件下,证明了算法的全局收敛性.数值结果表明,算法是有效的.     

Truncated regularized Newton method for convex minimizations

Recently, Li et al. (Comput. Optim. Appl. 26:131–147, 2004) proposed a regularized Newton method for convex minimization problems. The method retains local quadratic convergence property without requirement of the singularity of the Hessian. In this paper, we develop a truncated regularized Newton method and show its global convergence. We also establish a local quadratic convergence theorem for the truncated method under the same conditions as those in Li et al. (Comput. Optim. Appl. 26:131–147, 2004). At last, we test the proposed method through numerical experiments and compare its performance with the regularized Newton method. The results show that the truncated method outperforms the regularized Newton method. The work was supported by the 973 project granted 2004CB719402 and the NSF project of China granted 10471036.     

有奇异解的无约束最优化问题的一种混合法

本文研究求解含有奇异解的无约束最优化问题算法 .该类问题的一个重要特性是目标函数的Hessian阵可能处处奇异 .我们提出求解该类问题的一种梯度 -正则化牛顿型混合算法 .并在一定的条件下得到了算法的全局收敛性 .而且 ,经一定迭代步后 ,算法还原为正则化 Newton法 .因而 ,算法具有局部二次收敛性 .     

一维凸函数牛顿法的全局收敛性及其应用

牛顿法是求解非线性方程F（x)＝0的一种经典方法。在一般假设条件下，牛顿法只具有局部收敛性。本文证明了一维凸函数牛顿法的全局收敛性，并且给出了它在全局优化积分水平集方法中的应用。     

On a result by Dennis and Schnabel for Newton’s method: Further improvements

We improve local convergence results for Newton’s method by defining a more precise domain where the Newton iterates lie than in earlier studies using Dennis and Schnabel-type techniques. A numerical example is presented to show that the new convergence radii are larger and new error bounds are more precise than the earlier ones.     

局部误差的有界约束非线性方程组的非单调信赖域算法

本文提供了在没有非奇异假设的条件下,求解有界约束半光滑方程组的投影信赖域算法.基于一个正则化子问题,求得类牛顿步,进而求得投影牛顿步.在合理的假设条件下,证明了算法不仅具有整体收敛性而且保持超线性收敛速率.     

A minimum effort optimal control problem for the wave equation

A minimum effort optimal control problem for the undamped wave equation is considered which involves L ^∞-control costs. Since the problem is non-differentiable a regularized problem is introduced. Uniqueness of the solution of the regularized problem is proven and the convergence of the regularized solutions is analyzed. Further, a semi-smooth Newton method is formulated to solve the regularized problems and its superlinear convergence is shown. Thereby special attention has to be paid to the well-posedness of the Newton iteration. Numerical examples confirm the theoretical results.     

An operator Newton method for the Stefan problem based on smoothing: A local perspective

The initial/Neumann boundary-value enthalpy formulation for the two-phase Stefan problem is regularized by smoothing. Known estimates predict a convergence rate of ^1/2, and this result is extended in this paper to include the case of a (nonzero) residual in the regularized problem. A modified Newton Kantorovich framework is established, whereby the exact solution of the regularized problem is replaced by one Newton iteration. It is shown that a consistent theory requires measure-theoretic hypotheses on the starting guess and the Newton iterate, otherwise residual decrease is not expected. The circle closes in one spatial dimension, where it is shown that the residual decrease of Newton's method correlates precisely with the ^1/2 convergence theory.     

On a new family of high‐order iterative methods for the matrix pth root

The main goal of this paper is to approximate the principal pth root of a matrix by using a family of high‐order iterative methods. We analyse the semi‐local convergence and the speed of convergence of these methods. Concerning stability, it is well known that even the simplified Newton method is unstable. Despite it, we present stable versions of our family of algorithms. We test numerically the methods: we check the numerical robustness and stability by considering matrices that are close to be singular and badly conditioned. We find algorithms of the family with better numerical behavior than the Newton and the Halley methods. These two algorithms are basically the iterative methods proposed in the literature to solve this problem. Copyright © 2015 John Wiley & Sons, Ltd.