Multiple populations co-evolutionary particle swarm optimization for multi-objective cardinality constrained portfolio optimization problem
Introduction
Portfolio optimization problem (POP) aims to improve the portfolio return and reduce the portfolio risk in the complex financial market [1], which has gained increasing interests in recent years [2], [3], [4], [5]. To be more precise, POP can be regarded as how to allocate asset to help the investors obtain maximize return and minimize risk in a portfolio. As the financial market rapidly changes, POP becomes an important reference for investors to determine the effective portfolio from thousands of stocks. Although the POP is performed on historical data to find out the effective portfolio, the result is still promising to guide/help investors quickly make a suitable investment decision (i.e., select the suitable stocks to obtain maximize return and minimize risk) on the new-faced financial market. Therefore, researches into POP have significant reference meaning for investors to make quickly investment decision and achieve the goals of increasing returns and reducing risks [6], [7], [8], [9], [10].
Generally speaking, as the POP is from the practical application in financial market, there are different investment constraints such as transaction cost, bounds on holdings, and cardinality constrained (CC) that should be considered when deal with POP. Specially, the POP with CC (termed as CCPOP) is to limit the number of invested stocks to a certain value. The CCPOP is reasonable for most of the investors because they always pay attention to a limited number of stocks rather than all the stocks, and therefore the CCPOP has been widely regarded as a typical POP model. As the CCPOP has been proved to be a NP-Complete problem [11], many researchers tried to use evolutionary computation (EC) algorithms to solve it, including the genetic algorithm (GA) [12], differential evolution (DE) [13], [14], artificial bee colony [15], particle swarm optimization (PSO) [16], [17], and ant colony optimization [18]. In addition, some researchers utilized neural network [19] or statistics approach [20] to deal with CCPOP. Recently, Tuba et al. [21] combined the artificial bee colony with the firefly search (ABC-FS) to solve CCPOP. Chang et al. [22] employed PSO, tabu search, and simulate anneal to solve CCPOP. Yang et al. [23] converted the CCPOP to multi-portfolio optimization and borrowed the concept of Nash Equilibrium to deal with it.
In fact, the key to CCPOP is how to satisfy the CC. The common methods of dealing with CC mainly rely on some classical techniques, such as penalty function, repair method, and randomly selection. Streichert et al. [24] adopted a repair operation to satisfy CC, which removed some stocks with the small asset weights from the portfolio. Fieldsend et al. [25] designed a two-dimensional CC to better deal with CCPOP, which had advantages in some particular cardinality. Pai et al. [26] proposed a k-means clustering method by using the mean and variance of returns to satisfy CC. Although tremendous efforts have been put into solving CCPOP, how to satisfy the CC to obtain the suitable portfolio from thousands of stocks is still a challenging issue. Especially, as the stock market changes rapidly, the long computational time caused by penalty, repair, and randomly selection methods may be impractically and therefore a fast CC handling method is in great need.
Moreover, with respect to each stock, the high portfolio return generally means high portfolio risk, how to balance the portfolio return and risk is another key issue in CCPOP. However, the return and risk of stocks are usually conflict with each other. Therefore, how to achieve a reasonable trade-off between the portfolio return and risk is still difficult.
Subsequently, many researchers transformed CCPOP into the multi-objective CCPOP (MoCCPOP) and utilized the multi-objective evolutionary algorithms (MOEAs) to deal with MoCCPOP. Metawa et al. [27] proposed a self-organizing method for loan portfolio optimization by using GA, whose objectives were maximum the bank profit and minimum the bank loan. To enhance the algorithm performance in dealing with MoCCPOP, Qu et al. [28] proposed a pre-selection (PS) method to design a normalized MOEA based on decomposition (NMOEAD), resulting in the NMOEAD-PS algorithm. Chen et al. [29] proposed a MOEA based on local search, which combined the non-dominated sorting and a local search schema (NSLS) [30] to deal with MoCCPOP. The NMOEAD-PS and NSLS are two well-performing MOEAs in solving MoCCPOP.
Recently, many co-evolutionary MOEAs are proposed to better deal with the multi-objective optimization problems (MOP) [31], [32], [33], [34], [35], [36], [37]. Zhan et al. [31] the first time proposed a coevolutionary technique based on multiple populations for multiple objectives (MPMO) framework for solving MOPs. Later, the MPMO framework has become a new and efficient paradigm for solving MOPs, which has obtained great success in many real-world problems [32], [33], [34], [35]. Tian et al. [36] proposed a co-evolutionary constrained multi-objective optimization (CCMO) framework for constrained MOPs. Liang et al. [37] proposed a novel decomposition-based multi-objective co-evolutionary algorithm that used subpopulations to enhance each objective. Inspired by the endocrine regulation mechanism, Yao et al. [32] proposed an endocrine-based PSO (EBPSO) algorithm for solving the multi-objective workflow scheduling in cloud computing platform. These algorithms all use the co-evolutionary technique to solve the MOPs and obtain promising results. Therefore, this paper also focuses the co-evolutionary technique to deal with the MoCCPOP.
To deal with the CC challenge and the multi-objective challenge in POP, an efficient co-evolutionary MOEA named multiple populations co-evolutionary PSO (MPCoPSO) algorithm is proposed in this paper based on the efficient co-evolutionary MPMO [31] framework to solve the MoCCPOP. In addition, four novel strategies are incorporated into the proposed MPCoPSO algorithm, which are summarized as follows.
- 1)
A hybrid binary and real (HBR) encoding strategy is proposed to determine which stocks are selected and how many assets are allocated to each selected stock. The HBR encoding strategy uses two strings (i.e., binary string and real value string) to satisfy the CC encoding requirement in MoCCPOP. Firstly, binary coding 0/1 string is adopted to represent the selection of each stock. Secondly, the allocated asset of each selected stock is continuous value and is represented by real coding string.
- 2)
A return risk ratio heuristic (R3H) constraint handling strategy based on the historical return and risk of each stock is proposed to quickly select the suitable stocks for a promising portfolio, helping the proposed algorithm adapt to the rapidly changing stock market.
- 3)
A bi-directional local search (BLS) strategy is designed, which generates two new promising solutions in bi-directions around the current solution based on Gaussian distribution to assist the position update of particles. The BLS strategy can help improve the solution accuracy for approaching the global Pareto front (PF) in the MoCCPOP.
- 4)
A hybrid elite competition (HEC) strategy is proposed to update the archive by collecting all possible elite solutions generated during the current generation and then competitively preserving the solutions from these elites that are suitable for our MoCCPOP. The HEC strategy helps increase the diversity of solutions for exploring more Pareto optimal solutions in the MoCCPOP.
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. The experiments have been conducted not only on the widely used test data in OR-library [22], but also on the real-world data come from China’s Shenzhen stock market numbered 000001-002200. To validity the performance of MPCoPSO, we compare it with three well-performing algorithms designed for solving MoCCPOP, three widely used state-of-the-art MOEAs, and three recent well-performing MOEAs, the results show that our MPCoPSO is promising in solving MoCCPOP.
The rest of the paper is organized as follows. Section 2 introduces the basic PSO algorithm and POP, CCPOP, and MoCCPOP models. Section 3 shows the proposed MPCoPSO algorithm for solving the MoCCPOP model. Next, Section 4 presents the experimental results and makes discussion and analysis. Finally, Section 5 gives the conclusions.
Section snippets
PSO
PSO was proposed by Kennedy and Eberhart in 1995 [38], derived from the simulation of a simplified social model. The inertia weight is introduced to better control the exploitation and exploration state, forming a standard version. Subsequently, Zhan et al. introduced an adaptive PSO [39] and a discrete PSO [40] to improve the performance and practicability of PSO. Zeng et al. proposed the switching delayed PSO algorithm to solve the real-world problems [41], [42]. Recently, Zhang et al. [43]
MPCoPSO algorithm
To describe the MPCoPSO algorithm more clearly, the HBR encoding strategy and the R3H constraints process strategy for handling CC are first introduced. Then, the method of fitness evaluation is introduced. Later, the MPMO framework is described, followed by the particle update based BLS strategy and the archive update based HEC strategy being introduced in MPCoPSO. At last, the complete MPCoPSO algorithm is presented.
Experimental settings
The experimental data is divided into two datasets with five test cases in each dataset. The first dataset comes from the widely used dataset in OR-library, which can be downloaded from http://people.brunel. acuk~mastjjb/jeb/orlib/portinfo.html. The basic information of the first dataset is shown in Table 1. The second dataset comes from China’s Shenzhen stock numbered 000001–002200. The daily stock price changes of five test cases are calculated, which are 2016–12, 2017–09, 2017–10, 2017–11,
Conclusion
In this paper, a novel MPCoPSO algorithm for MoCCPOP is proposed based on the MPMO framework. Two populations are adopted to optimize the portfolio return and risk objectives, respectively. The proposed HBR encoding strategy and the R3H constraint handling strategy can provide a fast CC handling method to make our algorithm more suitable in the rapidly changing stock market. Besides, two novel strategies of the particle update based BLS strategy and the archive update based HEC strategy are
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
This work was supported in part by the National Key Research and Development Program of China under Grant 2019YFB2102102, in part by the Outstanding Youth Science Foundation under Grant 61822602, in part by the National Natural Science Foundations of China (NSFC) under Grant 61772207 and Grant 61873097, in part by the Key-Area Research and Development of Guangdong Province under Grant 2020B010166002, in part by the Guangdong Natural Science Foundation Research Team under Grant 2018B030312003,
Hong Zhao received the Bachelor’s degree in computer science from Henan Polytechnic University, Jiaozuo, China, in 2014, and the Master’s degree in computer science from Kunming University of Science and Technology, Kunming, China, in 2017, respectively. She is currently working toward the Ph.D. degree in South China University of Technology, Guangzhou, China. Her current research interests include artificial intelligence, evolutionary computation, swarm intelligence, and their applications in
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Hong Zhao received the Bachelor’s degree in computer science from Henan Polytechnic University, Jiaozuo, China, in 2014, and the Master’s degree in computer science from Kunming University of Science and Technology, Kunming, China, in 2017, respectively. She is currently working toward the Ph.D. degree in South China University of Technology, Guangzhou, China. Her current research interests include artificial intelligence, evolutionary computation, swarm intelligence, and their applications in design and optimization.
Zong-Gan Chen received the B.S. degree in 2016 from Sun Yat-Sen University, Guangzhou, China. He is currently pursuing the Ph.D. degree in computer science and technology at South China University of Technology, Guangzhou, China.
His current research interests include differential evolution, ant colony optimization, estimation of distribution algorithm, and their applications in real-world optimization problems.
Zhi-Hui Zhan (IEEE Senior Member) received the Bachelor’s degree and the Ph. D degree in Computer Science from the Sun Yat-Sen University, Guangzhou China, in 2007 and 2013, respectively. He is currently the Changjiang Scholar Young Professor with the School of Computer Science and Engineering, South China University of Technology, Guangzhou, China. His current research interests include evolutionary computation algorithms, swarm intelligence algorithms, and their applications in real-world problems, and in environments of cloud computing and big data. Dr. Zhan’s doctoral dissertation was awarded the IEEE Computational Intelligence Society (CIS) Outstanding Ph. D. Dissertation and the China Computer Federation (CCF) Outstanding Ph. D. Dissertation. He was a recipient of the Outstanding Youth Science Foundation from National Natural Science Foundations of China (NSFC) in 2018 and the Wu Wen-Jun Artificial Intelligence Excellent Youth from the Chinese Association for Artificial Intelligence in 2017. Dr. Zhan is listed as one of the Most Cited Chinese Researchers in Computer Science. He is currently an Associate Editor of the IEEE Transactions on Evolutionary Computation, the Neurocomputing, and the International Journal of Swarm Intelligence Research.
Sam Kwong (IEEE Fellow) received his BSc degree and MASc degree in electrical engineering from the State University of New York at Buffalo, USA and University of Waterloo, Canada, in 1983 and 1985 respectively. In 1996, he later obtained his PhD from the University of Hagen, Germany. From 1985 to 1987, he was a diagnostic engineer with the Control Data Canada where he designed the diagnostic software to detect the manufacture faults of the VLSI chips in the Cyber 430 machines. He later joined the Bell Northern Research Canada as a Member of Scientific staff. In 1990, he joined the City University of Hong Kong as a lecturer in the Department of Electronic Engineering. He is currently a Professor in the department of computer Science. His research areas are in Pattern Recognition, Evolutionary Computations and Video Analytics. Prof. Kwong is the Vice President of IEEE Systems, Man and Cybernetics (SMC). He was appointed as IEEE Distinguished Lecturer for the IEEE SMC society in 2017. He is currently an Associate Editor of the IEEE Transactions on Evolutionary Computation.
Jun Zhang (IEEE Fellow) received the Ph.D. degree in Electrical Engineering from the City University of Hong Kong, in 2002. He is currently a visiting scholar with Victoria University, Melbourne, Australia. His current research interests include computational intelligence, cloud computing, high performance computing, operations research, and power electronic circuits. Dr. Zhang was a recipient of the Changjiang Chair Professor from the Ministry of Education, China, in 2013, the China National Funds for Distinguished Young Scientists from the National Natural Science Foundation of China in 2011 and the First-Grade Award in Natural Science Research from the Ministry of Education, China, in 2009. He is currently an Associate Editor of the IEEE Transactions on Evolutionary Computation, the IEEE Transactions on Cybernetics, and the IEEE Transactions on Industrial Electronics.