Research On Sparse Portfolio Selection Models And Their Algorithms

The sparse portfolio selection problem has always been one of the central and active topics of the research on modern finance. Studying and solving this issue usually relies on the construction of a cross-disciplinary platform with the comprehensive application of a variety of disciplines such as matrix theory, economics, operation research and system engineering as well for the sake of abundant research achievements.Based on Markowitz’s portfolio selection theory methods that rational investors adopt for asset portfolio and the 1/2L regularization theory, this thesis studies two kinds of sparse portfolio selection models, improves and develops two algorithms including the successive over relaxation Half(SOR-Half) threshold algorithm and symmetric successive over relaxation Half(SSOR-Half) threshold algorithm to solve this two kinds of models, and obtains some results. More specifically, three aspects of research results as follows:Firstly, two kinds of sparse portfolio selection models are proposed to study the sparse portfolio selection problem including the sparse index tracking problem based on the least squares regression model and the quantile regression model.Secondly, the SOR-Half threshold algorithm and the SSOR-Half threshold algorithm are proposed to solve sparse portfolio selection model with least square regression based on the 1/2L regularization theory and the Half threshold algorithm, and then, the convergence of the two algorithms are well proved. Numerical experiments show that the algorithms proposed are more efficient. On the other hand, we also present a new Half threshold algorithm to solve sparse quantile portfolio selection model and prove its convergence.Finally, these algorithms proposed above are applied to the sparse portfolio selection models with least square regression and quantile regression, respectively. Numerical experiments and simulations demonstrate that: i) the SSOR-Half threshold algorithm is more efficient than Lasso algorithm and Half threshold algorithm; and ii) the Half threshold algorithm is superior to Lasso algorithm when solving the sparse quantile portfolio selection model.