文档介绍：摘 要
信赖域算法是非线性优化中一类重要的数值计算方法。它首先由 Powell 于 1970 年提出,随后在不断的研究中,人们发现了其诸多优点, 例如可靠性强,能解决 Hessian 矩阵不正定,迭代点收敛到鞍点等问题, 并且具有全局收敛性。因此信赖域算法和线搜索算法并列为求解非线性规划问题的两类主要的数值计算方法。
论文主要研究求解无约束最优化的非单调信赖域算法。将近年来基于函数平均权重非单调技术与信赖域算法相结合提出了三种新的非单调信赖域算法,证明了新算法的全局收敛性并用数值试验证明了算法的有效性。
全文共分为五章。第一章首先介绍了最优化算法的基本理论和无约束最优化算法的发展历史,然后回顾了信赖域算法的研究状况,指出了发展前景。第二章讨论了一般的信赖域算法的基本思想和基本结构,在此基础上引入了非单调信赖域算法。第三章构建了一类新的非单调线搜索信赖域算法,即在试探步失败时不重解信赖域子问题,而采用非单调线搜索技术来得到新的迭代点,节省了迭代步数,并给出了其全局收敛性的证明。第四章构建了一类非单调自适应信赖域算法,即在每次迭代时充分利用当前迭代点所包含的一次导数的信息来自动产生一个信赖域半径,并给出了其全局收敛性的证明。第五章构建了一类带线搜索的非单调自适应信赖域算法即结合前两章的改进方法,最后证明了其全局收敛性。并用数值试验验证了新算法的有效性。
关键词无约束最优化;信赖域算法;非单调;线搜索;全局收敛性
Abstract
Trust region algorithm is a class of important pute method of nonlinear optimization. It first proposed by Powell in 1970. Then people discover its many advantage in the continual studying. It have better reliability; for example ,it can resolve Hessian negative definite, convergence to saddle eatl problems; and retains the global convergence. So trust region algorithm and linear search method are tow important pute method for solving nonlinear programming.
Thesis main study unconstrained optimzation nonmonotone trust region bining the based on repeated function average value step down nonmonotonic technology which proposed in recent years and trust region algorithm. Propose three new nonmonotone trust region algorithm, and respectively proved the global convergence of the new last, numerical experiment prove the validity of the algorithm.
The full thesis is divided into four chapters. The first chapter introduced the optimization basic theory and unconstrained optimization development history.
Then retrospect research condition of trust region algorithm. Pointed out the development prospects of trust region algorithm. The second chapter,
discussed the general trust region algorithm basic idea and the basic structure. Based on this, introduction the nonmonotone trust region algorithm. The third chapter construct a nonmonotone trust