A Bundle Method With Inexact Data For CvaR Portfolio Optimization Problem

It is one of the important research topics in economics to make portfolio investments on selected risk assets and how to minimize the investment risk under the given income expectation.Conditional Value at Risk(CVa R)has become the mainstream tool to measure financial risk owing to its excellent nature.In this paper,a CVa R model of portfolio optimization problem is analyzed by a proximal bundle method with inexact data.Firstly,a piecewise linear approximation model of the objective function is constructed under the condition of using the function with inexact value.Utilizing the approach of bundle method,the corresponding subproblem is presented.Then,the original subproblems are transformed into a series of quadratic programming subproblems with the help of conventional bundle methods.Moreover,the dual subproblem is analyzed,and some derived results are obtained based on the correlation between the two optimal solutions.In the second part,the bundle method with inexact data is presented.Finally,the convergence of the proposed method is analyzed theoretically in detail.The sequence generated by the constructed algorithm converges to the optimal solution of the original problem,even though the data we used is inexact.In the first chapter,the research background of the CVa R portfolio optimization problem is expounded.Through simple analysis,it is concluded that the CVa R model for solving the portfolio optimization problem is a kind of non-smooth optimization problems.Then several methods for solving non-smooth optimization problems are introduced,including the subgradient method,cutting-planes method,exact bundle method,and inexact bundle method.Subsequently,the preliminary knowledge and relevant conclusions of the above several kinds of bundle methods are introduced,which lays a solid theoretical foundation for model construction,algorithm construction,and convergence analysis of the proposed algorithm.The second chapter focuses on the basic ideas of solving CVa R portfolio optimization problems.Firstly,a convex piecewise linear model of the objective function in the CVa R portfolio optimization problem is constructed by using the cutting-planes model,and then a quadratic programming penalty subproblem which produces the next trial point is constructed by using the approach of bundle method.Then,the expression of the optimal solution and the related properties of the subproblem are studied and analyzed.In chapter 3,on the theoretical basis of the first two chapters,this chapter puts forward a kind of inexact bundle method based on the CVa R portfolio optimization problem and gives a detailed explanation of the parameter setting and specific operation steps in the algorithm.In chapter 4,the convergence of the algorithm given in the third chapter is discussed respectively in the two cases where the algorithm produces an infinite number of descent steps and a finite number of descent steps(followed by an infinite number of null steps),and the satisfactory convergence results are obtained.