The Research On The Augmented Lagrangian Approach To Some No-short-sale Portfolio Selection Problems

Portfolio selection is to provide quantitative analysis to financial assets and formulate a mathematical model in order to find the highest return of a given risk level and make a investment strategy under certain analytical framework. No-short-sale-mean-variance portfolio selection problem is a very important subject of financial mathematics. The investor set up proper mathematical model to get the optimum strategy with the help of the statistical method and the capability of data processing of the computer, facing the fluctuation of the financial market, when short sale is forbidden. The no-short-sale-mean-variance portfolio selection model has no display solution, there are a lot of optimization algorithms which can help to give a solution to the model. In this paper, we discuss the application of the augmented lagrange function method in a series of no-short-sale-mean-variance portfolio selection problems.In the introduction of this paper, we give a short description of the background and present situation of the portfolio. In the second chapter, we discuss how the investor of different risk-tolerance makes a optimum portfolio strategy to get the highest return using augmented lagrange method, then we give an example. In the following part, we do more detailed analysis, we use max-min principle to find the optimum investment strategy in the worst situation under the framework of the mean-variance analysis, then we use the augmented lagrange method of the second chapter to solve the problem and give some numerical example. In the fourth chapter, we consider the fiction factor of the market, we introduce some friction factor, such as tax and transaction costs. We discuss how the investor make optimum investment decision with the help of the max-min model, then give a numerical example.