A bi-level optimization approach for portfolio problems with cardinality constraints

The portfolio problem is one of the most important problems in the financial field. The bi-level programming model of the problem, which is composed of Sharpe ratio (SR) model and mean-variance (MV) model, can effectively evaluate the relationship between risk and return, and obtain favorable investment strategy. However, the cardinality constraints make the portfolio problems more complex than ever, and the existing traditional methods are almost never efficient for the kind of problem involving both integer variables and hierarchical structure. Focusing on the shortcomings, this manuscript proposes a bi-level optimization method for portfolio problems with cardinality constraints based on evolutionary algorithms (EAs). In the proposed algorithm, an approximate evaluation strategy is presented and used to deal with the simplified lower level problem, which can efficiently save computational costs caused by obtaining lower level optimal solutions. In addition, the crossover and mutation operators are developed to improve the convergence and diversity of the populations. Experiments are carried out on five real data sets with different scales to verify the effectiveness of the proposed algorithm, and the computational results show the proposed algorithm obtains the better portfolio strategy than compared studies.