解非线性互补问题的非单调非精确Broyden-like算法

本文通过构造一个新的光滑互补函数,将非线性互补问题等价转换为光滑方程组问题.将非单调线搜索技术与非精确Broyden-like算法相结合,建立了解非线性互补问题的非单调非精确Broyden-like算法.在一定条件下证明了该算法的全局收敛性和局部二次收敛性.数值实验表明该算法对求解非线性互补问题是十分有效的.     

非线性均衡问题一个超线性收敛的光滑逼近SQP算法

研究非线性均衡问题,引入一个磨光算子将原问题转化为光滑问题,并用此光滑问题来逼近原来的问题而求解.在每步迭代中,通过转轴运算,求解一个线性约束二次规划问题和显式修正方向来得到主方向,并通过一个显式公式来得到高阶修正方向使得算法避免Maratos效应.在不需要上层互补条件下证明了算法具有全局收敛性和强收敛性且具有超线性收敛速度.     

非线性最小二乘问题的信赖域方法

本文提出一个新的非线性最小二乘的信赖域方法,在该方法中每个信赖域子问题只需要一次求解,而且每次迭代的一维搜索步长因子是给定的,避开一维搜索的环节,大大地提高了算法效率.文中证明了在一定的条件下算法的全局收敛性.     

求解非线性互补问题的一类光滑Broyden-like方法

非线性互补问题可以等价地转换为光滑方程组来求解. 基于一种新的非单调线搜索准则, 提出了求解非线性互补问题等价光滑方程组的一类新的非单调光滑 Broyden-like 算法.在适当的假设条件下, 证明了该算法的全局收敛性与局部超线性收敛性. 数值实验表明所提出的算法是有效的.     

求解广义非线性互补问题的光滑化拟牛顿法

利用光滑对称扰动Fischer-Burmeister函数将广义非线性互补问题转化为非线性方程组,提出新的光滑化拟牛顿法求解该方程组.然后证明该算法是全局收敛的,且在一定条件下证明该算法具有局部超线性(二次)收敛性.最后用数值实验验证了该算法的有效性.     

一个解凸二次规划的预测-校正光滑化方法

本文为凸二次规划问题提出一个光滑型方法,它是Engelke和Kanzow提出的解线性规划的光滑化算法的推广。其主要思想是将二次规划的最优性K-T条件写成一个非线性非光滑方程组,并利用Newton型方法来解其光滑近似。本文的方法是预测-校正方法。在较弱的条件下,证明了算法的全局收敛性和超线性收敛性。     

有界约束半光滑方程组的非单调投影信赖域方法

投影信赖域策略结合非单调线搜索算法解有界约束非线性半光滑方程组.基于简单有界约束的非线性优化问题构建信赖域子问题,半光滑类牛顿步在可行域投影得到投影牛顿的试探步,获得新的搜索方向,结合非单调线搜索技术得到回代步,获得新的步长.在合理的条件下,证明算法不仅具有整体收敛性且保持超线性收敛速率.引入非单调技术能克服高度非线性的病态问题,加速收敛性进程,得到超线性收敛速率.     

非线性互补问题光滑化拟牛顿算法的收敛性分析

给出求解p_0函数非线性互补问题光滑化拟牛顿算法,在p_0函数非线性互补问题有非空有界解集且F'是Lipschitz连续的条件下,证明了算法的全局收敛性.全局收敛性的主要特征是不需要提前假设水平集是有界的.     

基于一个新的NCP函数的光滑牛顿法求解非线性互补问题

本文研究了非线性互补的光滑化问题.利用一个新的光滑NCP函数将非线性互补问题转化为等价的光滑方程组,并在此基础上建立了求解P0-函数非线性互补问题的一个完全光滑化牛顿法,获得了算法的全局收敛性和局部二次收敛性的结果.并给出数值实验验证了理论分析的正确性.     

求解一般非线性互补问题的光滑化方法

在利用Fischer-Burmeister函数将非线性互补问题转化为非线性方程组的基础上,本文通过将信赖域方法与线性搜索方法结合起来,提出了求解一般非线性互补问题的光滑化方法.算法中我们给出了一个特定条件,条件满足时,采用信赖步,条件不满足时.采用梯度步.我们证明了算法具有全局收敛性.在解是R-正则的条件下,收敛速度是Q-超线性/Q-二阶收敛的.     

On iterative methods for solving nonlinear least squares problems over convex sets

Nonlinear least squares problems over convex sets inR ⁿ are treated here by iterative methods which extend the classical Newton, gradient and steepest descent methods and the methods studied recently by Pereyra and the author. Applications are given to nonlinear least squares problems under linear constraint, and to linear and nonlinear inequalities. Part of the research underlying this report was undertaken for the Office of Naval Research, Contract Nonr-1228(10), Project NR047-021, and for the U.S. Army Research Office — Durham, Contract No. DA-31-124-ARO-D-322 at Northwestern University. Reproduction of this paper in whole or in part is permitted for any purpose of the United States Government.     

Bernstein–Frechet inequalities for the parameter of the first order autoregressive process

The autoregressive process takes an important part in predicting problems leading to decision making. In practice, we use the least squares method to estimate the parameter of the autoregressive process. In the case of the first order autoregressive process, we know that the least squares estimator converges in probability to the unknown parameter θ. In this Note, we show that the least squares estimator converges almost completely to θ and so we construct the inequalities of type Bernstein–Frechet for the coefficient of the first order autoregressive process. Using these inequalities a confidence interval is then obtained. To cite this article: A. Dahmani, M. Tari, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Nonlinear conjugate gradient methods with structured secant condition for nonlinear least squares problems

In this paper, we deal with conjugate gradient methods for solving nonlinear least squares problems. Several Newton-like methods have been studied for solving nonlinear least squares problems, which include the Gauss-Newton method, the Levenberg-Marquardt method and the structured quasi-Newton methods. On the other hand, conjugate gradient methods are appealing for general large-scale nonlinear optimization problems. By combining the structured secant condition and the idea of Dai and Liao (2001) [20], the present paper proposes conjugate gradient methods that make use of the structure of the Hessian of the objective function of nonlinear least squares problems. The proposed methods are shown to be globally convergent under some assumptions. Finally, some numerical results are given.     

Some applications of polynomial optimization in operations research and real-time decision making

We demonstrate applications of algebraic techniques that optimize and certify polynomial inequalities to problems of interest in the operations research and transportation engineering communities. Three problems are considered: (1) wireless coverage of targeted geographical regions with guaranteed signal quality and minimum transmission power, (2) computing real-time certificates of collision avoidance for a simple model of an unmanned vehicle (UV) navigating through a cluttered environment, and (3) designing a nonlinear hovering controller for a quadrotor UV, which has recently been used for load transportation. On our smaller-scale applications, we apply the sum of squares (SOS) relaxation and solve the underlying problems with semidefinite programming. On the larger-scale or real-time applications, we use our recently introduced “SDSOS Optimization” techniques which result in second order cone programs. To the best of our knowledge, this is the first study of real-time applications of sum of squares techniques in optimization and control. No knowledge in dynamics and control is assumed from the reader.     

Local convergence theorems of Newton's method for nonlinear equations using outer or generalized inverses

We provide local convergence theorems for Newton's method in Banach space using outer or generalized inverses. In contrast to earlier results we use hypotheses on the second instead of the first Fréchet-derivative. This way our convergence balls differ from earlier ones. In fact we show that with a simple numerical example that our convergence ball contains earlier ones. This way we have a wider choice of initial guesses than before. Our results can be used to solve undetermined systems, nonlinear least squares problems and ill-posed nonlinear operator equations.     

Merit functions for general mixed quasi-variational inequalities

In this paper, we present some merit functions for general mixed quasi-variational inequalities, and we obtain the equivalent optimization problems to general mixed quasi-variational inequalities. Since the general mixed quasi-variational inequalities include general variational inequalities, quasi-variational inequalities and nonlinear (implicit) complementarity problems as special cases, our results continue to hold for these problems. In this respect, results obtained in this paper represent an extension of previously known results.     

A NEW SECANT METHOD FOR NONLINEAR LEAST SQUARES PROBLEMS

This paper is concerned with the solution of the nonlinear least squares problems. A new secant method is suggested in this paper, which is based on an affine model of the objective function and updates the first-order approximation each step when the iterations proceed. We present an algorithm which combines the new secant method with Gauss-Newton method for general nonlinear least squares problems. Furthermore, we prove that this algorithm is Q-superlinearly convergent for large residual problems under mild conditions.     

Semilocal Convergence Theorems for Newton’s Method Using Outer Inverses and Hypotheses on the Second Fréchet-Derivative

 We provide semilocal convergence theorems for Newton’s method in Banach space using outer or generalized inverses. In contrast to earlier results we use hypotheses on the second instead of the first Fréchet-derivative. This way our Newton-Kantorovich hypotheses differ from earlier ones. Our results can be used to solve undetermined systems, nonlinear least squares problems and ill-posed nonlinear operator equations. We complete our study with some very simple examples to show that our results apply, where others fail.     

Semilocal Convergence Theorems for Newton’s Method Using Outer Inverses and Hypotheses on the Second Fréchet-Derivative

 We provide semilocal convergence theorems for Newton’s method in Banach space using outer or generalized inverses. In contrast to earlier results we use hypotheses on the second instead of the first Fréchet-derivative. This way our Newton-Kantorovich hypotheses differ from earlier ones. Our results can be used to solve undetermined systems, nonlinear least squares problems and ill-posed nonlinear operator equations. We complete our study with some very simple examples to show that our results apply, where others fail. (Received 26 April 2000; in final form 17 November 2000)     

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.