文档介绍：变分不等式问题的信赖域方法
摘要
信赖域方法是一种求解无约束优化问题比较有效的方法,然而信赖域半径的选取对
,利用当前迭代
点的梯度和海赛阵信息来选取当前迭代步的信赖域半径,数值试验表明,该方法比一般
的信赖域方法有效。本文结合当前比较流行的非单调技术,提出了一类改进的非单调自
适应信赖域方法,这样可以减少计算量。当试探步长不可接受时不重解信赖域子问题而
采用非单调线搜索技术. 在一般假设条件下,不仅可以说明至少存在一个聚点为稳定点,
而且有该改进的算法所产生的点列的任何聚点都是原问题的稳定点。另外,此改进的算
法仍保持超线性收敛性。
本文不仅对变分不等式进行了线性化而且对约束集进行线性化,把约束集为一类特殊
的非空闭凸集的变分不等式转化为等价的约束集为多面凸集的线性变分不等式,再利用投
影收缩方法进行计算。
本文将箱约束变分不等式转化为等价的无约束最优化问题,再利用信赖域方法求解。
关键词:信赖域方法,非单调自适应技术,线搜索技术,箱约束变分不等式,D-间隙函数
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南京航空航天大学硕士学位论文
Abstract
Trust region method is a kind of efficient and robust method to solve general unconstrained
optimization. But the choice of the trust region radius is very important to the efficiency of the
method. Recently Zhang proposed a new trust region method called adaptive trust region
method, in which the trust region radius depends on the gradient and Hesse matrix information
of the current iterate point. Numerical results show the efficiency of the method. This thesis
tombines the nonmonotone technique to the adaptive trust region method, adaptive
nonmonotonic trust region algorithm for unconstrained optimization is improved, it takes
nonmonotonic linear search instead of resolving the subproblem when the trail step is not
essful. This may allow a putational saving. Due to particularity of trust
region’s radius, the paper is not only proved that at least a accumulation point is stability but also
that any accumulation point produced by this algorithm is stability for unconstrained
optimization. In addition, global convergence is proved under certain conditions.
This thesis is not only to linearize variational inequality but also linearize set of constraints.
Constraints to put a special kind of non-empty closed convex variational inequalities into the
equivalent of a muli-faceted set of constraints of linear convex variational inequalities, we
calculate it using projection and contraction method.
This paper convert boxed v