双稳定束方法以及收敛性分析

Doubly stabilized bundle method and its convergence

对于带有非线性约束的非光滑优化问题,束方法是最常用且最有效的方法之一。在目前众多束方法中,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,在数值计算中更加具有优势,而且具有很高的理论研究价值。主要研究双稳定性束方法及其收敛性。首先将双稳定束方法的子问题在新范数意义下应用对偶思想进行求解,得到与原范数意义下求解相类似的结果。接下来在已经求得新范数意义下解的基础上,对算法收敛性做进一步分析,即在一般迫近束方法算法的框架下讨论收敛性。假设算法不终止,无论产生无限多下降步,还是有限多下降步,不仅得到迭代序列的相应收敛结果,同时也得到了与单纯用迫近束方法求解无约束优化问题相类似的性质。

For nonsmooth optimization problems with nonlinear constraints, one of the most commonly used and the most effective methods is the bundle method. In the present bundle methods, the doubly stabilized bundle method is a new algorithm with high theory 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 try to study the doubly stabilized bundle method and its convergence. Firstly, in application of dual idea under the new norm sense, we solve the sub-problem of the doubly stabilized bundle method and get the similar results with the original norm sense. Then we study further the convergence of the doubly stabilized bundle method in the framework of general bundle algorithm, on the basis of solutions which we have got from solving the sub-problem of the doubly stabilized bundle method under the new norm sense. Assume that the algorithm is never stopped, and no matter there exist an infinite number or a finite number of descent steps, we not only obtain the corresponding results of the convergence of the iterative sequence, but also have similar properties to the case in which only the proximal bundle method is employed to solve the unconstrained optimization problems.