【作者】 李娜；

【导师】 沈洁；

【作者基本信息】 辽宁师范大学 ， 运筹学与控制论， 2015， 硕士

【摘要】 对于带有非线性约束的非光滑优化问题,双稳定束方法是结合迫近束方法与水平束方法产生的一种新算法,具有较高的理论和实际研究价值。本文对双稳定束方法的对偶子问题和算法的收敛性方面做了进一步研究,在一般束方法的算法框架下我们讨论了子问题的对偶表达和算法的一般收敛性条件,不仅得到算法产生的迭代序列的最终收敛性结果,同时还得到了与单纯使用迫近束方法求解无约束优化问题相类似的有关性质。论文主要分为以下几部分内容。第一章,主要介绍文中涉及到的非光滑优化的基本概念和几种常用的束方法,指出这几种束方法模型自身和它们的解之间的联系,为引出双稳定性束方法做准备。第二章,这一章是文章的主体部分。主要先简要介绍一下双稳定束方法的基本框架,然后对双稳定束方法子问题中目标函数的近似模型进行调整修正,使其子问题变为带有新范数的全新子问题。接下来再从对偶的角度研究子问题解的具体解析形式及相关性质,得到与原文献相吻合和类似的结论,即对偶子问题的解与之前迭代点的次梯度的凸组合有关,此外还有一些与算法参数相关的衍生结论也一并给出。第三章,这一章也是文章的主体部分。主要是在前一章的基础上,运用双稳定束方法子问题在新范数意义下产生的迭代序列,对其进行在一般束方法算法框架下的收敛性分析。为证明双稳定束方法在这样一个基本框架下的收敛性,假设算法从不停止,那么就有两种情况出现,或者生成无限多的下降步迭代点,或者生成了最后一个下降步迭代点。在新范数意义下对其进行收敛性分析,必然引入新的参量,那么在这两种情况下,通过很好的调整控制参量,在一定条件下都可以得到迭代序列收敛于问题最优值点的结论。更多还原

【Abstract】 For non-smooth optimization problems with nonlinear constraints,the doubly stabilized bundle method is a new algorithm with high theoretical and practical research value which combines the level bundle method with the proximal bundle method,and it has an obvious advantage in numerical computation.In this paper,we further study the dual sub-problem and convergence analysis of the doubly stabilized bundle method.Under the framework of general bundle algorithm,we discuss the expression of the dual sub-problem and the general convergence conditions of the algorithm,and not only obtain the results of convergence of the iterative sequences produced by the algorithm,but also have similar properties to the case in which only the proximal bundle method is employed to solve the unconstrained optimization problems.This paper is mainly divided into the following several parts.The first chapter,it mainly introduces the basic concepts of non-smooth optimization and several commonly used bundle methods involved in this paper.It also points out the relations between these bundle methods and their solutions,which make preparations for the doubly stabilized bundle method introduced in the next chapter.The second chapter,this chapter is the main part of the paper.Firstly,it briefly explains the basic framework of the doubly stabilized bundle method,and then adjusts the approximation model of the objective function of the doubly stabilized bundle method so that the sub-problem becomes a new one with a different norm.Then we study the concrete form and the corresponding properties of the solution of the sub-problem from the viewpoint of duality,and find that the form of the solution is quite similar to the original documents,and come to the conclusion that the solution of sub-problem is related to the convex combinations of the subgradient of the previous iteration,in addition some conclusions related with algorithm parameters are also given.The third chapter,this chapter is also the main part of the paper.Mainly on the basis of the previous chapter,we study the convergence of the doubly stabilized bundle method in the framework of general bundle algorithm by employing the iterative sequence which we have got from solving the sub-problems of the doubly stabilized bundle method under the new norm sense.To prove the convergence of the doubly stabilized bundle method under such a basic framework,assumes that the algorithm never stops,then there are two cases to be considered,either generate an infinite sequence of descent iterations,or generate the last iteration of the descent step.To analyze the convergence of the algorithm under the new norm sense,it is necessary to introduce some new parameters.In both cases,by adjusting parameters appropriately,we can obtain the conclusion that the iterative sequence converges to an optimal solution under certain conditions.更多还原