Doubly Stabilized Bundle Method And Its Convergence Analysis

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.