【作者】 周亚群；

【导师】 韦增欣；

【作者基本信息】 广西大学 ， 运筹学与控制论， 2007， 硕士

【摘要】 信赖域方法是近二十年来发展起来的一类重要的数值计算方法。由于具有很好的可靠性、强适性，以及很强的收敛性，目前它和传统的的线搜索方法并列为求解非线性规划的两类主要的数值计算方法。本文主要研究一个修改的BFGS公式在信赖域方法中的应用，其结构如下：第一章，回顾了信赖域算法的基本思想和研究状况，根据韦增欣等给出的新的拟牛顿方程，给出了一个新的BFGS校正公式，并分析了相关性质。第二章，结合新的校正公式，我们提出一种求解无约束优化问题的非单调的BFGS信赖域方法，并证明该方法求解非凸极小化问题的全局收敛性。该算法的优点是信赖域子问题的目标函数是一个严格凸二次函数，因而信赖域子问题的求解相对容易。而且，我们在不假设迭代矩阵序列有界的前提下建立算法的全局收敛性定理。第三章，将修改的BFGS公式与Armijo线搜索相结合，得到一个新的信赖域算法。在适当条件下，证明了该算法具有全局收敛性和超线性收敛性。数值结果表明此算法对无约束优化问题是有效的。更多还原

【Abstract】 Trust region method is an important numerical method developed over the last twodecades. Because of its reliability, robustness and strong convergence, trust region method,together with line search method, has become one of the two main numerical methods forsolving nonlinear programming. We mainly discuss the application of a modified BFGSformula to trust region method. The thesis is organized as follows:In Chapter 1, we review the basic thought and research of the trust region method.According to the new quasi Newton equation proposed by Wei at al., we give a new modifiedBFGS formula and analyze the corresponding properties.In Chapter 2, corresponding to the new formula, we develop a nonmonotone BFGStrust-region method for solving unconstrained optimization problems. A good property of themethod is that the objective function of the trust-region subproblem is a strictly convexquadratic function. Consequently, it is relatively easy to solve the subproblem. We establish aglobal convergence theorem for the method without requirement of the boundedness of thegenerated matrix sequence.In Chapter 3, combining the modified BFGS formula with Armijo line search, wepropose a new trust region method, and prove that the algorithms are global and superlinearconvergence under some mild conditions. Preliminary numerical results show that the methodis efficient and advisable for nonlinear optimization.更多还原