基于四阶拟牛顿方程的信赖域算法研究

【作者】 王晓明；

【导师】 王希云；

【作者基本信息】 太原科技大学 ， 应用数学， 2012， 硕士

【摘要】 拟牛顿算法是牛顿法的一种推广，牛顿法在每一次迭代过程中都需要很大的工作量来计算海森阵,而且海森阵不好计算，甚至可以说很难求.为了克服牛顿法的这一缺点，研究者以拟牛顿方程为基础构造了拟牛顿算法.利用目标函数的一阶导数信息，构造目标函数的近似曲率，这种算法不需计算海森阵并且在一定的条件下具有超线性收敛性.在拟牛顿算法中拟牛顿方程具有至关重要的作用.传统的拟牛顿方程忽略了目标函数值的信息只是利用它的梯度信息.考虑到充分利用信息资源，许多研究人员对传统的拟牛顿方程进行了修正，在此基础上提出了新的拟牛顿算法.本文主要是基于文献[7]提出的四阶拟牛顿方程，结合信赖域算法、线搜索技术、非单调技术、自适应技术，针对二次模型及新锥模型进行了研究，提出了关于二次模型及新锥模型的非单调信赖域算法和非单调自适应信赖域算法.这些基于拟牛顿方程的混合算法不仅能保留原有拟牛顿算法的大多数良好性质，并且在近似目标函数的二次曲率时比拟牛顿算法具有更高的精度.本文的主要研究的内容如下：第一、对于二次模型的研究，在四阶拟牛顿方程及其修改的BFGS校正公式的基础上，将非单调技术、自适应技术与Armijo线搜索、信赖域算法相结合提出了两类非单调自适应信赖域算法．在一定的条件下证明了算法的收敛性.并且给出了相应的数值实验结果.第二、对于新锥模型的研究，利用四阶拟牛顿方程及其修改的BFGS校正公式，将非单调技术与Wolfe线搜索、信赖域算法相结合提出了一类拟牛顿非单调信赖域算法.在较弱的条件下，证明了此算法的全局收敛性.数值结果表明该算法是有效的.第三、对于新锥模型的研究，，在四阶拟牛顿方程及其修改的BFGS校正公式的基础上，将非单调技术、自适应技术与Armijo线搜索、信赖域算法相结合提出了两类关于新锥模型的非单调自适应信赖域算法，在一定的条件下证明了算法的收敛性.并且给出了相应的数值实验结果.更多还原

【Abstract】 Quasi-Newton algorithm is the promotion of the Newton method. It needsmuch of work in each iteration of using the Newton method to calculate theHessian matrix. By using Newton method and it is not easy or even difficult tocalculate the Hessian matrix. In order to overcome this disadvantage of Newtonmethod, the researchers constructed a method named quasi-Newton algorithmbased on quasi-Newton equation. By using the information of the first derivativeof the objective function, quasi-Newton constructs the approximate curvature ofthe objective function. This algorithm don＇t need to calculate to the Hessianmatrix and has super linear convergence.The quasi-Newton algorithm plays a crucial role in unconstrainedoptimization process. Traditional quasi-Newton equation ignores theinformation of the objective function value; it only uses its gradient information,it is a great waste of information resources. Taking the full use of informationresources into account, at present many researchers modified the traditionalquasi-Newton equation and proposed a new quasi-Newton algorithm. In thispaper, combining with the trust region algorithm, line search technologies, takesa research on the second model and cone model based on the literature [7] inwhich put forward four-order quasi-Newton equation. These mixed algorithmbasing on the quasi-Newton algorithm not only retain the majority properties ofthe old quasi-Newton algorithm, but also can be more exact than the old one inquadratic curvature approximate of the objective function. The main contentsof this paper are as follows:Firstly, on the basis of the new quasi-Newton equation with four order anda BFGS corrector formula, we introduce a concept of self-adapt. According toArmijo line search technique, a new non-monotonic trust region algorithm isproposed. In the paper, the convergence of the algorithm is proved in certaincondition, and the relevant numerical result is given at the same time.Secondly, on the basis of the new quasi-Newton equation with four order and a BFGS corrector formula, we propose a quasi-Newton non-monotonic trustregion algorithm combining both the non-monotonic Wolfe line searchtechnique and trust region method about the new cone model. In lowercondition, we prove that the new algorithm is convergence. And the numericalresult shows the algorithm is effective.Thirdly, on the basis of the new quasi-Newton equation with four order anda BFGS corrector formula, we propose a non-monotonic trust region algorithmabout the new cone model, and the convergence of the algorithm in certaincondition is proved, and the relevant numerical result is given at the same time.更多还原