Multiple populations co-evolutionary particle swarm optimization for multi-objective cardinality constrained portfolio optimization problem

With the rapid development of financial market, a growing number of stocks become available on the financial market. How to efficiently select these stocks to achieve higher return and lower risk has become a hot research topic in financial management. This is usually called the portfolio optimization problem (POP). When the cardinality constrained (CC) is added to limit the number of selected stocks to a certain value, the resulting CCPOP is more challenging with the following two difficulties: i) Due to the complexity of CC in finical market, how to efficiently deal with CC in POP to obtain feasible solution is difficult and time-consuming. ii) The objectives of portfolio return and risk always conflict with each other and their relation is difficult to balance. To better deal with above difficulties, this paper focuses on the multi-objective CCPOP (MoCCPOP) and proposes a multiple populations co-evolutionary particle swarm optimization (MPCoPSO) algorithm, which is based on multiple populations for multiple objectives (MPMO) framework and has the following four advantages. Firstly, a hybrid binary and real (HBR) encoding strategy is introduced to better represent the stock selection and the asset weight of the solutions in MoCCPOP. Secondly, a return risk ratio heuristic (R3H) strategy based on the historical return and risk of each stock is proposed as a fast CC handling method to obtain feasible solutions. Thirdly, a new particle update method based on bi-directional local search (BLS) strategy is designed to increase the chance to improve the solution accuracy and to approach the global Pareto front (PF). Last but not least, a hybrid elite competition (HEC) strategy is proposed to assist the archive update, which provides more promising solutions and brings diversity to avoid local PF. The first two strategies help to efficiently deal with the CC challenge, while the last two strategies are efficient in solving the multi-objective challenge. By comparing with some recent well-performing and state-of-the-art multi-objective optimization algorithms, MPCoPSO shows the superior performance in solving the MoCCPOP.