非线性互补约束问题一个全局收敛的SQP算法

本文研究非线性互补约束优化问题,利用Fischer-Burmeister函数将非线性互补问题转化为非光滑方程,提出一个求解非线性互补约束问题的SQP算法,并在适当的假设下证明这个算法是全局收敛的.     

非线性规划的一个全局收敛的一般算法模式及应用

非线性规划的一个全局收敛的一般算法模式及应用赵云彬，李俭（中国科学院应用数学研究所，北京１０００８０）（重庆大学数学系，重庆６３００４４）考虑约束问题中Ｊ：＊”－－＋Ｒ；ｈ：Ｈ”、＊ｍ且０＜。＜＋．Ｘ分别为如下三种约束：（１）ｘ一＊”Ｚ（１１）ｘ一｛...     

具有全局收敛性的求解对称非线性方程组的一个修改的信赖域方法

本文给出一个求解非线性对称方程组问题的修改的信赖域方法,在适当的条件下我们将建立此方法的全局收敛性.对给定的问题而言,数值结果表明此方法是有效的.     

解凸约束非线性单调方程组的无导数谱PRP投影算法

本文在著名PRP共轭梯度算法的基础上研究了一种无导数谱PRP投影算法,并证明了算法在求解带有凸约束条件的非线性单调方程组问题的全局收敛性.由于无导数和储存量小的特性,它更适应于求解大规模非光滑的非线性单调方程组问题.数值试验表明,新算法对给定的测试问题是有效的和稳定的.     

求解互补问题的Gauss-Newton方法的全局收敛性

求解互补问题的Gauss-Newton方法是由Subramamian提出的．本文研究了此方法的收敛性质，在较弱的情况下，建立了一个全局收敛结果，此结果是相关文献中的结果的推广．     

二阶非线性微分方程零解的全局渐近稳定性

本文研究了具有四个非线性项零解的全局渐近稳定性。     

非线性约束优化问题的一个新的全局收敛算法

对于非线性约束的优化问题.最近给出的各种SQP算法均采用罚函数技巧以保证算法的全局收敛性,因而都必须小心地调整惩罚参数。本文给出一个不依赖于惩罚参数、每步迭代的校正矩阵也不需正定且仍具有全局收敛性的SQP方法,而且罚函数形式简单、具有和约束函数同阶的光滑性.     

一类二阶非线性常微分方程零解的全局稳定性

本文用变量分离法研究了二阶非线性微分方程ｘ＋ φ（ｘ）ｐ（ｘ ）＋ ｇ（ｘ）ｆ （ｘ）＝ ０ 零解的全局稳定性     

一类二阶非线性微分方程解的全局渐近稳定性

给出了Ｌｉｅｎａｒｄ方程零解全局渐近稳定的一个充分必要条件,并进一步研究了方程ｘ＋ｆ（ｘ）ｘ＋ｇ（ｘ）＝ｅ（ｔ）解的渐近性态与方程ｘ＋ｆ（ｘ）＋ｇ（ｘ）＝ｅ（ｔ;ｘ,ｘ）零解的全局渐近稳定性。     

求全局最优解的新算法

利用局部极大值点与动力系统的稳定奇点的对应性，计算代数方程的根、无约束极大值点、有约束极大值点、非线性规划解、及最小二乘解．我们采用了常微分方程数值解的Euler算法及网格初始点的循序迭代算法，并以具体的例子和程序说明创立的方法具有通用性，同时考虑了一些存在的问题以便在理论和算法上作进一步的改进。     

The Lagrangian Globalization Method for Nonsmooth Constrained Equations

The difficulty suffered in optimization-based algorithms for the solution of nonlinear equations lies in that the traditional methods for solving the optimization problem have been mainly concerned with finding a stationary point or a local minimizer of the underlying optimization problem, which is not necessarily a solution of the equations. One method to overcome this difficulty is the Lagrangian globalization (LG for simplicity) method. This paper extends the LG method to nonsmooth equations with bound constraints. The absolute system of equations is introduced. A so-called Projected Generalized-Gradient Direction (PGGD) is constructed and proved to be a descent direction of the reformulated nonsmooth optimization problem. This projected approach keeps the feasibility of the iterates. The convergence of the new algorithm is established by specializing the PGGD. Numerical tests are given. This author's work was done when she was visiting The Hong Kong Polytechnic University. His work is also supported by the Research Grant Council of Hong Kong.     

NCP Functions Applied to Lagrangian Globalization for the Nonlinear Complementarity Problem

Based on NCP functions, we present a Lagrangian globalization (LG) algorithm model for solving the nonlinear complementarity problem. In particular, this algorithm model does not depend on some specific NCP function. Under several theoretical assumptions on NCP functions we prove that the algorithm model is well-defined and globally convergent. Several NCP functions applicable to the LG-method are analyzed in details and shown to satisfy these assumptions. Furthermore, we identify not only the properties of NCP functions which enable them to be used in the LG method but also their properties which enable the strict complementarity condition to be removed from the convergence conditions of the LG method. Moreover, we construct a new NCP function which possesses some favourable properties.     

两类解非线性约束问题的超记忆可行方向法

本文将无约束超记忆梯度法推广到非线性不等式约束优化问题上来,给出了两类形式很一般的超记忆可行方向法,并在非退化及连续可微等较弱的假设下证明了其全局收敛性.适当选取算法中的参量及记忆方向,不仅可得到一些已知的方法及新方法,而且还可能加快算法的收敛速度.     

Lagrangian Globalization Methods for Nonlinear Complementarity Problems

This paper extends the Lagrangian globalization (LG) method to the nonsmooth equation arising from a nonlinear complementarity problem (NCP) and presents a descent algorithm for the LG phase. The aim of this paper is not to present a new method for solving the NCP, but to find such that when the NCP has a solution and is a stationary point but not a solution.     

Spectral DY-Type Projection Method for Nonlinear Monotone Systems of Equations

In this paper, we propose a spectral DY-type projection method for nonlinear monotone systems of equations, which is a reasonable combination of DY conjugate gradient method, the spectral gradient method and the projection technique. Without the differentiability assumption on the system of equations, we establish the global convergence of the proposed method, which does not rely on any merit function. Furthermore, this method is derivative-free and so is very suitable to solve large-scale nonlinear monotone systems. The preliminary numerical results show the feasibility and effectiveness of the proposed method.     

求解约束规划的一个非线性Lagrange函数

本文提出了一个求解不等式约束优化问题的非线性Lagrange函数,并构造了基于该函数的对偶算法.证明了当参数σ小于某一阈值σ_0时,由算法生成的原始-对偶点列是局部收敛的,并给出了原始-对偶解的误差估计.此外,建立了基于该函数的对偶理论.最后给出了算法的数值结果.     

An Liu-Storey-Type Method for Solving Large-Scale Nonlinear Monotone Equations

An algorithm for solving nonlinear monotone equations is proposed, which combines a modified Liu-Storey conjugate gradient method with hyperplane projection method. Under mild conditions, the global convergence of the proposed method is established with a suitable line search method. The method can be applied to solve large-scale problems for its lower storage requirement. Numerical results indicate that our method is efficient.     

Globalization of Newton's Method for Solving Non-linear Equations

The Lagrangian globalization (LG) method for non-linear equation-solving proposed in [10] is developed through theoretical analysis, the formulation of a particular LG algorithm, and a numerical illustration. New merit functions (termed detour potentials) for non-linear equation-solving, which broaden the LG concept, are also defined.     

On the Convergence of a Trust-Region Method for Solving Constrained Nonlinear Equations with Degenerate Solutions

This paper presents a trust-region method for solving the constrained nonlinear equation F(x) = 0, x , where R ⁿ is a nonempty and closed convex set, F is defined on the open set containing and is continuously differentiable. The iterates generated by the method are feasible. The method is globally and quadratically convergent under local error bounded assumption on F. The results obtained are extensions of the work of Yamashita Fukushima (Ref. 1) and Fan Yuan (Ref. 2) for unconstrained nonlinear equations. Numerical results show that the new algorithm works quite well.     

约束优化问题的广义既约型梯度法

本文提出了二类解约束优化问题的广义既约型梯度法,从统一角度研究了投影梯度法和既约梯度法的结构及其全局收敛性.本文结果统一、推广了常见的可行方向法.