An Improved Hybrid Bundle Method For Solving A Class Of Portfolio Optimization Problems

The bundle method is well known as one of the most robust and promising algorithms for solving non-smooth optimization problems^[21].Both theory and practice have shown that the bundle method can combine descent with stability,and it has been widely used in many fields.The applications of the bundle method to solve non-smooth optimization problems is highly efficient.The characteristics of this method are as follows.It can establish an information bundle to retain the obtained iteration information,and can memorize the“best”iteration point obtained so far.In each iteration,this"best"point is kept in the bundle,and on this basis,we will continue to search for the approximate optimal solution to the considered problem until the optimal solution of the problem is finally found.Firstly,this paper studies the optimization problems based on the CVaR model.After constructing the subproblem which produces the next trial point,the dual theory will be used to give an explicit representation of the solution to the original sub-problem and the dual sub-problem.At the same time,the relevant important relationships are obtained.These conclusions are vital to the construction of the algorithm and to the analysis of the convergence of the algorithm.Based on the above discussion,combining the characteristics of the problem itself,and further combining the proximal bundle method with the level bundle method,a hybrid bundle method for solving a class of portfolio optimization problems is proposed.This method introduces the non-Euclidean distance--Bregman distance,which forms a more comprehensive sub-problem.Finally,a detailed theoretical analysis of the convergence of the proposed level bundle method is carried out.Under the framework of the general bundle method,the algorithm still has a good convergence result.This article is divided into three parts,the overall structure is as follows.In the first chapter,the research background and current situation of portfolio optimization problems are introduced and we also introduce various bundle methods for solving non-smooth optimization problems in detail.These methods include trust region bundle method,proximal bundle method and level bundle method.In order to make readers better understand the overall framework of this paper,this chapter gives the relevant preparatory knowledge and related theoretical results,which lays a theoretical foundation for the further study.In the second chapter,the basic idea of the general bundle method for solving the optimization problem of the CVaR model is firstly introduced.The construction of the approximate sub-problems of the CVaR model is studied,and the dual theory is used to give explicit expressions for the solutions to the original sub-problem and the dual sub-problem.It gives some important conclusions related to the subgradient and approximate subgradient.At the same time,the basic idea of the hybrid bundle method is given to pave the way for the next chapter to study the hybrid bundle method for solving the CVaR portfolio optimization problem.The third chapter focuses on the new bundle method for the CVaR portfolio optimization problem.This method combines the proximal bundle method with the level bundle method,and uses Bregman distance as the distance measurement method.Based on the general bundle method framework,a non-smooth improved hybrid bundle method for solving the CVaR portfolio optimization problem is given.The convergence of the algorithm is analyzed in detail,and the discussion is mainly divided into three cases.In the first case,the level set D_k=?appears infinitely many times.In the second case,the algorithm generates finite descent steps.In the third case,the algorithm produces infinite descent steps.According to the above three cases,the convergence of the new bundle method for the CVaR portfolio optimization problem is studied.