A Class Of Modified Accelerated Proximal Level Bundle Methods For Convex Optimization Problems

Convex optimization is an important branch of optimization,which is widely used in stochastic programming,engineering design,the optimal control and other fields.With the advent of the era of big data and the rise of machine learning,many convex optimization problems with special structure and large scale have emerged.Therefore,the study of effective algorithms for solving convex optimization problems have important scientific significance and practical application value.Firstly,a new method of accelerating proximal level bundle method is proposed to improve the method of accelerating proximal level bundle method in literature[Guanghui Lan,Mathematical Programming,149:1-45,2015].Lan's method needs to solve two subproblems in every iteration.Compared with Lan's method,the modified accelerated proximal level bundle method proposed in this paper only needs to solve one subproblem,which can reduce the computation of the algorithm.Compared with the classical level bundle method,the proposed method combines the multi-step accelerated strategy and introduces three iterative point sequences for the solution.The traditional Euclidean distance is extended by introducing prox-function,which can make full use of the geometric properties of the feasible set.By analyzing the complexity of the proposed modified accelerated proximal level bundle method,the uniform optimal iterative complexity can be obtained for smooth,weakly smooth and non-smooth convex optimization problems.Secondly,we introduce the modified uniform smoothing level bundle method base on uniform smoothing level bundle method of Lan for a class of structured nonsmooth convex optimization problems.This method changes solving two subproblems in Lan into solving one subproblem,which can reduce iterative computation.The objective function of structured nonsmooth optimization problem is composed of a nonsmooth convex function and a simple convex function,the proposed method for nonsmooth function to carry on the smoothing,and uses modified accelerated proximal level bundle method to solve the problem.This method can automatically adjust the smoothing parameter and achieve the same iterative complexity as Nesterov's smoothing method without input of any parameters of the problem.Finally,we report some preliminary numerical results for a class of modified accelerated proximal level bundle methods in this thesis,and the numerical results show that the proposed methods are effective and have certain advantages.