Barzilai-Borwein Methods For The Maximal Eigenvalue Problem Of High Dimensional Symmetric Matrices And The Numerical Method For The Two-stage Optimization Of Risk Profit

In this paper, we study numerical methods for two types of mathematical prob-lems. One is the largest eigenvalue problem of high dimensional symmetric matrices, the other is the numerical method of the two-stage risk-profit optimization of portfo-lio problems. For the first problem, we propose two Barzilai-Borwein-Like (BB-like) gradient methods, and two new algorithms for such eigenvalues problem. Further, we compare our methods with other methods. For the second problem, we establish the two-stage Worst-Case risk profit optimization model and design the corresponding computing method. When the random variable obey the ellipsoid distribution, we ap-ply our method to the power market based on Monte Carlo approach. The paper is organized as follows:The first chapter is introduction. We introduce back ground of our topic, current research situation and the outline of this paper.The second chapter introduce the model and algorithm for computing the largest eigenvalue of high dimensional symmetric matrices. First we give the details of the existed Barzilai-Borwein (BB) gradient methods, and put forward two new BB-like methods using novel steps and directions. Then, we propose a new Wolf line search method. Based on the new BB steps and Wolf line search direction. two new algorithms are presented. Finally. we prove the global convergence of the two methods and give a counter example for the second method. Some preliminary numerical results show the efficiency of the new algorithmsIn the third chapter. we study the two-stage Worst-Case optimization of risk-profit model and algorithm. We introduce the Worst-Case Value-at-risk (WCVaR) robust optimization model. A two-stage Worst-Case risk-profit optimization model is put forward when the random variable obey the ellipsoid distribution. Under the condition that the lost function is linear, we transform the model to linear programming by the Lagrange dual theory, and prove the equivalence of models. Finally, we apply our model to the power allocation problem of the generation company. Numerical examples display the efficiency of our algorithm.Some conclusions and further works were given in the last chapter.