非线性优化问题的一类非拟牛顿算法研究

【作者】 刘洪伟；

【导师】 焦宝聪；

【作者基本信息】 首都师范大学 ， 应用数学， 2004， 硕士

【摘要】 对于非线性优化问题寻找快速有效的算法一直是优化专家们研究的热门方向之一。文[28]基于校正的非拟牛顿方程，给出了无约束优化的一类非拟牛顿算法。本文结合文[28]中的非拟牛顿法，给出了求解无约束非线性优化问题的一类具有超线性收敛的非拟牛顿算法。 在第一章我们首先简要的介绍了最优化问题的提出以及判断最优解常用的最优性条件，回顾了无约束优化问题常用的几类导数下降类算法。 在第二章中，就非拟牛顿族在无约束最优化问题上，采用非单调线搜索下是否具有全局收敛性进行了研究。在目标函数满足一致凸的条件下，证明了非拟牛顿族是全局收敛的。 在第三章中，就非拟Newton族在无约束最优化问题上，采用非精确线搜索下是否具有全局收敛性进行了研究。文中提出了一种非拟Newton族校正，并证明在目标函数满足梯度Lipschitz连续的条件下，此校正在Wolfe或Armijo线搜索下具有全局收敛性。更多还原

【Abstract】 Seeking fast theoretical convergence and effective algorithms in unconstrained optimization is a very interested research topic for the optimization specialists and engineers. Paper [28] gives a class of non-quasi-Newton algorithms about unconstrained programming problems based on the modified non-quasi-Newton equation. In this paper, we give a class of superlinearly convengent algorithms for nonlinear programming problems with unconstrained by combining non-quasi-Newton methods [28] with some inexact line searches.In chapter 1 ,we first introduce the development of optimization and some extensive optimality conditions which to decide the optimum solution. We review several extensive derivative descent methods of unconstrained programming.In chapter 2,the non-quasi-Newton’s family is concerned with the problem of whether the method with inexact line search converges globally when applied to unconstrained optimization problems.We propose a update and prove that the method with either a Wolfe-type or an Armijo-type line search converges globally if the function to be minimized has Lipschitz continuous gradients.In chapter 3,the non-quasi-Newton Methods for unconstrained optimization is investigated.Non-monotone line search procedure is introduced, which is combined with the non-quasi-Newton family.Under the uniformly convexity assumption on objective function,the global convergence of the quasi-Newton family is proved.更多还原