同伦摄动法在求解四阶边值问题中的应用

将同伦摄动法用于求解常微分方程四阶边值问题.通过将常微分方程边值问题转化为积分方程组,应用同伦摄动法求得近似解.给出同伦摄动法在两个具体的实例中的应用,并将近似解与精确解进行了比较,验证了同伦摄动法对求解线性、非线性常微分方程边值问题是一种非常有效的方法.     

符号型Nagumo条件下一类非线性边值问题的可解性

综合利用Leray-Schauder度理论的同伦不变性、上下解方法等,在符号型Nagumo条件下获得了一类三阶非线性常微分方程在非线性边界条件下解的存在性结果.     

非线性刚性变延迟微分方程单支方法的数值稳定性

现有文献中对于非线性延迟微分方程渐近稳定性及其数值方法的稳定性研究大都局限于常延迟的情形,例如可参见匡蛟勋[1-3],黄乘明[4],Torelli[5]等人的大量工作.1994年A.Iserles[6] 首次研究了比例延迟微分方程数值方法的线性稳定性,随后有相当多的文献对比例延迟微分方程的各种数值方法的线性稳定性进行了讨论.1997年Zennaro[7]首次研究了非线性刚性变延迟微分方程的渐近稳定性,但该文中对于延迟量的限制十分苛刻,同时该文也首次研究了非线性刚性变延迟微分方程Runge-Kutta方法的非线性稳定性. 本文目的是试图在上述基础上进一步研究非线性刚性变延迟微分方程的渐近稳定性及其数值方法的稳定性.首先在第二节我们给出了非线性刚性变延迟微分方程模型问题（2.1）渐     

具有收缩表面的二阶流体驻点流动的级数解

研究具有收缩表面的边界层流动的解析解.通过相似变换,将偏微分方程简化为可用同伦分析法(HAM)求解的常微分方程.然后讨论了具有收缩表面的二维轴对称流动.     

无界区域非凸优化问题的组合同伦方法

考虑带有不等式约束的优化问题,对此问题建立组合同伦方程,给出同伦路径存在的一个条件,此条件不需要可行域满足法锥条件,获得了优化问题的K-K-T点.     

非线性系统稳定分析的特征函数法及其应用

本文引入一个特征函数,用于定量刻画非线性常微分方程的指数稳定性。与常用的Lyapunov方法相比,该方法简单、易用、而且易获得对一族范数（所有单调范数）皆成立的稳定性条件。所获结果推广了稳定理论中的一些著名结论,并应用于非线性连续神经网络的指数稳定性分析,推广和深化了[1-3]所获得的基本结论。     

求解多目标规划最小弱有效解的同伦内点方法

本文利用非线性规划中的组合同伦方法;给出了求解目标规划问题最小弱有效解的同伦内点方法,并证明了该方法是整体收敛的。     

求解初值问题的RKNd方法

本文基于一阶常微分方程所导出的二阶微分方程提出RKNd方法,其内级阶比传统RK方法高一阶.RKNd方法的阶条件由特殊Nystr(o)m树给出.在相同级数下,RKNd方法可达到的最高代数阶比传统的RK方法高.数值实验结果表明RKNd方法比同阶RK方法在计算效率上具有一定的优越性.     

时滞微分方程依照两种测度的稳定性

运用变异Liapunov方法，讨论了时滞微分方程依照两种测度的稳定性。借助于中间测度h^*(t,x），在未扰动系统为常微分方程的情形下，得到了关于时滞微分方程非一致和一致稳定性的判定定理。     

弱拟法锥条件下非凸优化问题的同伦算法

本文给出弱拟法锥条件的定义,并针对非线性组合同伦方程,得到在弱拟法锥条件下求解约束非凸优化问题的同伦内点算法.证明了该算法对于可行域的某个子集中几乎所有的点,同伦路径存在,并且同伦路径收敛于问题的K-K-T点,通过数值例子验证了该算法是有效的.     

解无约束最优化的基于锥模型的过滤集- 信赖域方法

锥模型优化方法是一类非二次模型优化方法, 它在每次迭代中比标准的二次模型方法含有更丰富的插值信息. Di 和Sun (1996) 提出了解无约束优化问题的锥模型信赖域方法. 本文根据Fletcher 和Leyffer (2002) 的过滤集技术的思想, 在Di 和Sun (1996) 工作的基础上, 提出了解无约束优化问题的基于锥模型的过滤集信赖域算法. 在适当的条件下, 我们证明了新算法的收敛性. 有限的数值试验结果表明新算法是有效的.     

解大型无约束优化的新的非单调简单模型BB-TR方法（英文）

The trust region(TR) method for optimization is a class of effective methods.The conic model can be regarded as a generalized quadratic model and it possesses the good convergence properties of the quadratic model near the minimizer.The Barzilai and Borwein(BB) gradient method is also an effective method,it can be used for solving large scale optimization problems to avoid the expensive computation and storage of matrices.In addition,the BB stepsize is easy to determine without large computational efforts.In this paper,based on the conic trust region framework,we employ the generalized BB stepsize,and propose a new nonmonotone adaptive trust region method based on simple conic model for large scale unconstrained optimization.Unlike traditional conic model,the Hessian approximation is an scalar matrix based on the generalized BB stepsize,which resulting a simple conic model.By adding the nonmonotone technique and adaptive technique to the simple conic model,the new method needs less storage location and converges faster.The global convergence of the algorithm is established under certain conditions.Numerical results indicate that the new method is effective and attractive for large scale unconstrained optimization problems.     

一种混合的HS-DY共轭梯度法

本文在HS方法和DY方法的基础上,综合两者的优势,提出了一种求解无约束优化问题的新的混合共轭梯度法.在Wolfe线搜索下,不需给定下降条件,证明了算法的全局收敛性.数值试验表明,新算法较之HS方法和PR方法更加有效.     

A Combination of Variational and Penalty Methods for Solving a Class of Fractional Optimal Control Problems

This paper develops an approximate method, based on the combination of epsilon penalty and variational methods, for solving a class of multidimensional fractional optimal control problems. The fractional derivative is in the Caputo sense. In the presented method, utilizing the epsilon method, the given optimal control problem transforms into an unconstrained optimization problem; then, the equivalent variational equality is derived for the given unconstrained problem. The variational equality is approximately solved by applying a spectral method.     

基于动力系统的线性不等式组的解法

本文提出了一种新的求解线性不等式组可行解的方法-基于动力系统的方法．假设线性不等式组的可行域为非空,在可行域的相对内域上建立一个非线性关系表达式,进而得到一个结构简单的动力系统模型．同时,定义了穿越方向。文章最后的数值实验结果表明此算法是有效的．     

求解线性不等式组的方法

本提出了一个新的求解线性不等式组可行解的方法－－无约束极值方法。通过在线性不等式组的非空可行域的相对内域上建立一个非线性极值问题，根据对偶关系，得到了一个对偶空间的无约束极值及原始，对偶变量之间的简单线性映射关系，这样将原来线性不等式组问题的求解转化为一个无约束极值问题。中主要讨论了求解无约束极值问题的共轭梯度算法。同时，在寻找不等式组可行解的过程中，定义了穿越方向，这样大大减少计算量。中最后数值实验结果表明此算法是有效的。     

Applying the Powell's Symmetrical Technique to Conjugate Gradient Methods with the Generalized Conjugacy Condition

This article proposes new conjugate gradient method for unconstrained optimization by applying the Powell symmetrical technique in a defined sense. Using the Wolfe line search conditions, the global convergence property of the method is also obtained based on the spectral analysis of the conjugate gradient iteration matrix and the Zoutendijk condition for steepest descent methods. Preliminary numerical results for a set of 86 unconstrained optimization test problems verify the performance of the algorithm and show that the Generalized Descent Symmetrical Hestenes-Stiefel algorithm is competitive with the Fletcher-Reeves (FR) and Polak-Ribiére-Polyak (PRP⁺) algorithms.     

A new modified nonmonotone adaptive trust region method for unconstrained optimization

In this paper, we present an adaptive trust region method for solving unconstrained optimization problems which combines nonmonotone technique with a new update rule for the trust region radius. At each iteration, our method can adjust the trust region radius of related subproblem. We construct a new ratio to adjust the next trust region radius which is different from the ratio in the traditional trust region methods. The global and superlinear convergence results of the method are established under reasonable assumptions. Numerical results show that the new method is efficient for unconstrained optimization problems.     

A new family of hybrid three-term conjugate gradient methods with applications in image restoration

In this paper, based on the hybrid conjugate gradient method and the convex combination technique, a new family of hybrid three-term conjugate gradient methods are proposed for solving unconstrained optimization. The conjugate parameter in the search direction is a hybrid of Dai-Yuan conjugate parameter and any one. The search direction then is the sum of the negative gradient direction and a convex combination in relation to the last search direction and the gradient at the previous iteration. Without choosing any specific conjugate parameters, we show that the search direction generated by the family always possesses the descent property independent of line search technique, and that it is globally convergent under usual assumptions and the weak Wolfe line search. To verify the effectiveness of the presented family, we further design a specific conjugate parameter, and perform medium-large-scale numerical experiments for smooth unconstrained optimization and image restoration problems. The numerical results show the encouraging efficiency and applicability of the proposed methods even compared with the state-of-the-art methods.

     

无约束多目标优化的记忆梯度法

基于无约束单目标记忆梯度法,本文提出了一种无约束多目标优化问题的记忆梯度法,并证明了算法在Armijo线性搜索下的收敛性。数据试验结果验证了该算法的有效性。