Abstract
Portfolio management is an important technology for reasonable investment, fund management, optimal asset allocation, and effective investment. Portfolio optimization problem (POP) has been recognized as an NPhard problem involving numerous objectives as well as constraints. Applications of evolutionary algorithms and swarm intelligence optimizers for resolving multiobjective POP (MOPOP) have attracted considerable attention of researchers, yet their solutions usually convert MOPOP to POP by means of weighted coefficient method. In this paper, a multiswarm multiobjective optimizer based on poptimality criteria called pMSMOEAs is proposed that tries to find all the Pareto optimal solutions by optimizing all objectives at the same time, rather than through the above transforming method. The proposed pMSMOEAs extended original multiple objective evolutionary algorithms (MOEAs) to cooperative mode through combining poptimality criteria and multiswarm strategy. Comparative experiments of pMSMOEAs and several MOEAs have been performed on six mathematical benchmark functions and two portfolio instances. Simulation results indicate that pMSMOEAs are superior for portfolio optimization problem to MOEAs when it comes to optimization accuracy as well as computation robustness.
1. Introduction
In real life, most optimization problems are multiobjective optimization problem (MOP). In MOP, numerous contradictory objectives which are subject to several certain constraints must be optimized at the same time. In general, the common approach to solve MOP is to find Pareto optimal set. And Pareto optimality theory indicates that if no vector makes , vector is Pareto optimality. Pareto optimal set could improve at least one objective without deteriorating other objectives. Currently, MOP has made progress in both theory and application [1, 2], yet there are still many challenges because of its complexity. As an effective method to solve MOP, MOEAs could obtain solutions with good diversity [3]. Consequently, they are widely found in various practical applications, such as power dispatching [4], financial management [5], electric machine designing [6], and spectrum allocation [7].
As a problemsolving technique, the main strengths of MOEAs are their fast convergence and efficient search ability. However, it is still a challenge to overcome the local convergence or balance diversity and convergence of swarms in MOP for the researchers. Many researchers have attempted to settle this issue with the ideas of optimality criterion and multiswarm strategy. Optimality criterion could find the best solution according to the different needs of the problem. Yet the conventional Pareto method does not have the advantage of such flexibility. Meanwhile, multiswarm strategy is prominent in improving diversity.
In koptimality employed in literature [8], the dominating solution could have an inferior performance on some particular objectives, which is acceptable to decision maker. The number of objectives is prescribed previously by means of the adjustment of k value. This is difficult for other methods. In literature [9], Optimality Criteria method is employed to seek the optimum through finding a solution that satisfies some prespecified criteria, which are postulated to the corresponding optimal result for the problem. However, only applying optimality criteria, multiobjective algorithms may suffer from excessive loss of diversity. Indeed, multiswarm strategy could restrict such rapid convergence and increase diversity effectively due to cooperation and exchange between swarms. In literatures [10–13], several MOEAs introduce a multiswarm strategy. These proposed algorithms contain multiple slave swarms, and the quantity of slave swarms is equal to that of objective functions. During evolution, every slave swarm is dedicated to optimizing a certain objective to discover its nondominated solutions. An improved particle swarm optimization (PSO) algorithm is proposed in literature [14]. It is shown that whole swarm is stochastically separated into some smallscale subswarms. The swarm is reorganized stochastically every R generations. In that case, the good information obtained by each subswarm could be exchanged. In hybrid multiswarm PSO algorithm shown in literature [15], PSO and differential evolution approach are employed during evolution. It is worth noting that all above algorithms utilize only one strategy to improve the algorithm and suffer from the balance of diversity and convergence, which have the potential for improvement.
In this paper, two classes of strategies combining MOEAs are adopted to solve MOP, poptimality criteria [3], and multiswarm strategy. Poptimality criteria are employed in selection operator during evolution. These criteria have the ability of determining the most feasible solutions among ones located in the same nondomination rank. Therefore, they have the potential to better the convergence of feasible solutions in late stages of evolution. Meanwhile, competition and cooperation techniques among subswarms are designed in multiswarm strategy. Distribution list and replacement list are devised, guaranteeing the interaction between swarms. Multiple swarms are utilized for optimizing objectives, and each separated subswarm employs the MOEAs combining poptimality criteria to find out all the nondominated solutions. Multiswarm strategy enables improving the diversity of feasible solutions and preventing local convergence. A multiswarm multiobjective optimizer combining the above strategies, called multiswarm multiobjective optimization evolutionary algorithms based on poptimality criteria (pMSMOEAs), is proposed. Several groups of experiment are conducted to evaluate the performance of pMSMOEAs and MOEAs.
Simultaneously, pMSMOEAs are employed to solve MOPOP. Markowitz puts forward the classic meanvariance (MV) portfolio model, which became the theoretical basis of modern portfolios [16, 17]. In Markowitz’s approach, the expected return of portfolio and the risk of portfolio (represented by mean and variance of assets, respectively) are described as two criteria of portfolio model. In the presence of the above two criteria, there is a tradeoff between risk and return. Since Markowitz proposed the MV theory, a large number of studies have been done to extend or modify the basic model in different directions [18–20]. However, the study of multiobjective model considering expected cost of assets is a little rare. In this paper, a threeobjective portfolio model considering expected return, expected cost, and risk is designed to solve MOPOP. In the proposed model, expected return is measured by mean of assets and should be maximized, risk is weighed by semivariance of assets and should be minimized, and expected cost is represented by Euclidean distance of weight vectors and should be minimized. Afterwards, it is optimized by pMSMOEAs.
The remainder of this paper is structured as follows. Section 2 presents three MOEAs. In Section 3, after an introduction to poptimality criteria and multiswarm strategy, pMSMOEAs in this study are pictured. In Section 4, performance of pMSMOEAs and MOEAs on six commonly used multiobjective benchmark functions will be shown. In Section 5, fulfillment of pMSMOEAs for MOPOP is presented. Finally, Section 6 summarizes the paper.
2. MultiObjective Optimization Evolutionary Algorithms
2.1. NSGAII Algorithm
NSGAII algorithm named by Deb et al. has been identified as a computationally fast MOEA. And because of its simplicity, excellent diversity, and convergence of feasible solutions, this algorithm is proved to be one of the most efficient MOEAs. It is prominent in two aspects: fast nondominated sorting for individuals and elitist selection [21]. The fast nondominated sorting depends on the indexes of nondominated rank as well as crowding distance. Further information about process of NSGAII algorithms could be seen in [22].
2.2. MODE Algorithm
MODE algorithm is a simple and powerful MOEA for MOP over a continuous domain. The outstanding advantages of MODE are its speed and robustness. MODE is mainly composed of three components: mutation, Paretobased evaluation, and selection. Among them, Paretobased evaluation is the same as NSGAII. In addition, each vector of the individual undergoes a mutation process with certain mutation probability . At last, there is a parameter in selection operator that could indicate the distance between a solution and its surrounding solution in objective space. Further information about the process of MODE algorithms could be seen in [23].
2.3. MOEA/D Algorithm
The main idea of MOEA/D algorithm lies in decomposing MOP into multiple scalar subproblems and optimizing them at the same time [24]. Every subproblem utilizes the information of its neighboring subproblems, which reduces its computational complexity of each generation. Scalarizing functions which are provided with uniformly distributed weight vectors is the fitness evaluation condition of MOEA/D [25]. In the computational experiments of this paper, the Tchebycheff function is employed to decompose. Further information about the process of MOEA/D algorithms could be seen in [24].
3. MultiSwarm MultiObjective Optimizer Based on pOptimality Criteria
3.1. Introduction of POptimality Criteria
Poptimality criteria named by Emiliano (2014) are a new kind of optimality criteria to solve MOP. These criteria have the ability of determining the most feasible solutions among the ones located in the same nondomination rank.
As described in [3], the criteria are defined as follows. Let be finite set of feasible solutions. A vector is an optimal solution if and , where is the probability that is better than other solutions from in terms of an objective function (, ). P_{i}(a) is calculated as follows:where S_{i}(a) is the rank of a according to the objective function f_{i} through the quicksort algorithm.
Thence, the aim is searching a vector that maximizes or equivalently.
If maximizes , the following formula should be maximized as well.
Inspired by the pnorm, the following function is considered:where , and p_{i}(a) is calculated as (1). Then, poptimality criteria could be defined as follows.
Let be the finite set of feasible solutions. A vector is optimal if and pfunction() pfunction(a).
Poptimality criteria have the ability of determining the most feasible solutions among ones located in the same nondomination rank. Section 3.3 will illustrate the evolution of MOEAs combining poptimality criteria.
3.2. Introduction of MultiSwarm Strategy
The above three algorithms introduced in Section 2 use the analogy of single swarm. However, only with combining poptimality criteria, the above multiobjective algorithms may suffer from excessive loss of diversity. Indeed, multiswarm strategy could restrict such rapid convergence and increase diversity effectively due to cooperation and exchange between swarms.
The linchpin of multiswarm strategy lies in two aspects: on the one hand, K individuals with superior performance are selected from subswarms to compose a distribution list. Meanwhile, each subswarm prepares a replacement list comprised of K individuals with inferior performance. K is predefined, and the individuals in distribution list and replacement list are selected by nondomination rank and crowding distance.
The whole swarm is evolved in the form of predetermined number subswarms, each of them performs the same multiobjective algorithm at early stage. After some predefined generations of evolution, the individuals in distribution list from one subswarm are sent to the adjacent subswarm to replace individuals in its replacement list. Taking this order, the last subswarm performs information transformation with first subswarm, as shown in Figure 1.
Figure 1 illustrates hierarchical interaction topologies among individuals. Ring is adopted in swarm level, and ring as well as star topologies are simultaneously employed in individual level. Different subswarms are arranged in a unidirectional ring. In other words, every subswarm could accept individuals in its distribution list from adjacent subswarm to replace individuals of its replacement list. The simplicity of the structure makes this interaction topology very fast in interaction [26].
3.3. Description of the Proposed pMSMOEAs
Multiswarm nondominated sorting genetic algorithm II based on poptimality criteria, named pMSNSGAII, is demonstrated in this part.
3.3.1. MSNSGAII Algorithm
At the beginning of pMSNSGAII algorithm, the maximum number of cycles (MCN), the number of swarm maximum cycles (SMCN), and the number of exchange of swarms (EN) are set. The algorithm will loop a predefined number of times (MCN). As an example, the following tth loop will be described.
In initialization, m subswarms () of N solutions are stochastically generated. Each solution has n realvalued variables. The control parameter of poptimality criteria p_{j}∈ [plb, pub] (1 ≤ j ≤ m) is determined for each subswarm (plb and pub are lower and upper limits of p_{j}, respectively). Values p_{1}, p_{2}, …, p_{m} are isometric, and p_{1 }= plb and p_{m} = pub. Solutions in subswarm are sorted and assigned a certain rank based on the nondomination sorting.
For subswarm , three operators, including selection, recombination, and mutation, are involved in the update process. The current subswarm is represented as parent swarm. Firstly, a group of q solutions is stochastically selected from , and then the solution with the least value of pfunction is determined. After repetitions, N solutions are selected. Secondly, the recombination operator and mutation operator in NSGAII are employed. Finally, the new subswarm of size N is regarded as the offspring swarm. The combination of parent and offspring swarms is carried out. Consequently, nondomination sorting and crowding distance are utilized to create new parent swarm .
After predefined generations SMCN, K individuals with superior performance are selected from subswarms into distribution list, as introduced in Section 3.2. And K individuals with inferior performance are selected from subswarms into replacement list. After EN times of information transformation among subswarms, all subswarms merge into one whole swarm. Thereafter, the whole swarm is no longer partitioned.
3.3.2. Flowchart
The general steps of pMSNSGAII algorithm are shown in Algorithm 1. And the flowchart is illustrated in Figure 2.

3.3.3. Other pMSMOEAs
MODE and MOEA/D based on multiswarm strategy and poptimality criteria (named pMSMODE algorithm and pMSMOEA/D algorithm, respectively) are similar to the above pMSNSGAII algorithm.
The pMSMODE algorithm and pMSMOEA/D algorithm also start by stochastically generating m initial subswarms. For pMSMODE algorithm (or pMSMOEA/D algorithm), before evolution operation, the superior individuals are chosen from each subswarm based on poptimality criteria and then are placed in the intermediate mating pool (or temporary elite population) for subsequent evolution operation. After some predefined generations of evolution, K individuals with superior (inferior) performance are selected for information transformation same as pMSNSGAII algorithm. After EN times of information transformation among subswarms, all subswarms merge into one whole swarm.
4. Test and Results
To completely evaluate the performance of above pMSMOEAs without bias against some certain selected problems, four twoobjective as well as two threeobjective benchmark functions are utilized. Formulas of those functions could be seen in Appendix.
4.1. Evaluation Method
For the purpose of facilitating the quantitative assessment of the performance of proposed algorithms, two performance metrics should be considered: convergence metric γ and diversity metric Δ. More detailed information about them could be seen in [27, 28].
4.2. Experimental Setting
Experiments have been executed with pMSNSGAII, pMSMODE, pMSMOEA/D, NSGAII, MODE, and MOEA/D. To contrast different algorithms with a fair time metric during experiment, population size is set as 200, and the number of function evaluations (FEs) is set as 40000. Each algorithm runs 10 times, using a PC Intel Core i57200U, 2.50 GHz CPU with 8GB of RAM.
For NSGAII, MODE, and MOEA/D, parameter settings are the same as original algorithms described in [22–24], respectively.
For pMSNSGAII, pMSMODE, and pMSMOEA/D, the whole swarm with 200 individuals is equally partitioned into 4 subswarms. And the lower and upper bounds of the poptimality criteria’s interval are set as 0.5 and 2.0, respectively (according to [3]). The number of swarm maximum cycles (SMCN) and the number of exchange of swarms (EN) are set as 20 and 10, respectively. The algorithm will loop 5 times (MCN). And the other setting is the same as original algorithms.
4.3. Results and Discussion
Test results of pMSMOEAs and MOEAs on six benchmark functions, including maximum, minimum, average, and standard deviation of the convergence metric () and the diversity metric () values, are listed in this part. Besides, CPU time is employed to measure the time complexity of the algorithms. To further demonstrate performance of pMSMOEAs, results of another multiswarm algorithm, multihive multiobjective artificial bee colony (M^{2}OABC) algorithm proposed in [26], on several of above benchmark functions, are also listed in the following. M^{2}OABC algorithm has been proved to be effective and robust with combining multihive strategy.
4.3.1. TwoObjective Functions
Tables 1–4 and Figures 3–6 show the optimization results of these algorithms for twoobjective functions. In Figures 3–6, the solid lines represent true Pareto front (PF), while the star spots stand for found nondominated solutions.
(a) PF obtained by pMSNSGAII on SCH2
(b) PF obtained by NSGAII on SCH2
(c) PF obtained by pMSMODE on SCH2
(d) PF obtained by MODE on SCH2
(e) PF obtained by p MSMOEA/D on SCH2
(f) PF obtained by MOEA/D on SCH2
(a) PF obtained by pMSNSGAII on ZDT3
(b) PF obtained by NSGAII on ZDT3
(c) PF obtained by pMSMODE on ZDT3
(d) PF obtained by MODE on ZDT3
(e) PF obtained by pMSMOEA/D on ZDT3
(f) PF obtained by MOEA/D on ZDT3
(g) PF obtained by M2OABC on ZDT3
(a) PF obtained by pMSNSGAII on ZDT4
(b) PF obtained by NSGAII on ZDT4
(c) PF obtained by pMSMODE on ZDT4
(d) PF obtained by MODE on ZDT4
(e) PF obtained by pMSMOEA/D on ZDT4
(f) PF obtained by MOEA/D on ZDT4
(a) PF obtained by pMSNSGAII on ZDT6
(b) PF obtained by NSGAII on ZDT6
(c) PF obtained by pMSMODE on ZDT6
(d) PF obtained by MODE on ZDT6
(e) PF obtained by pMSMOEA/D on ZDT6
(f) PF obtained by MOEA/D on ZDT6
(g) PF obtained by M2OABC on ZDT6
On SCH2 function, it can be noticed that pMSMOEAs have superior performance to MOEAs in terms of metric and metric after 40000 FEs from Table 1. Figure 3 shows that pMSMOEAs have great potential to discover a welldistributed as well as diverse solution set for SCH2 function. Yet MOEA/D only finds a sparse distribution, although it can basically archive true PF for SCH2.
On ZDT3, ZDT4, and ZDT6 functions, Tables 2–4 show that performance of pMSMOEAs in both γ metric and metric is better than that of MOEAs and lightly better than that of M^{2}OABC. Figures 4–6 show that MOEAs produce poor results on these test functions and they are almost impossible to achieve true PF, while pMSMOEAs have great potential to approach true PF.
4.3.2. ThreeObjective Functions
Figure 7 shows the true PF for two threeobjective functions. Tables 5 and 6 and Figures 8 and 9 show the optimization results of pMSMOEAs, MOEAs, and M^{2}OABC algorithms on DTLZ2 and DTLZ3.
(a) True PF on DTLZ2
(b) True PF on DTLZ3
(a) PF obtained by pMSNSGAII on DTLZ2
(b) PF obtained by NSGAII on DTLZ2
(c) PF obtained by pMSMODE on DTLZ2
(d) PF obtained by MODE on DTLZ2
(e) PF obtained by pMSMOEA/D on DTLZ2
(f) PF obtained by MOEA/D on DTLZ2
(a) PF obtained by pMSNSGAII on DTLZ3
(b) PF obtained by NSGAII on DTLZ3
(c) PF obtained by pMSMODE on DTLZ3
(d) PF obtained by MODE on DTLZ3
(e) PF obtained by pMSMOEA/D on DTLZ3
(f) PF obtained by MOEA/D on DTLZ3
On DTLZ2 function, when given 10000 FEs for seven algorithms, performance of pMSMOEAs is better than that of MOEAs and is comparable to that of M^{2}OABC as shown in Table 5. Figure 8 shows that pMSMOEAs have great potential to obtain a superior PF for DTLZ2, especially pMSNSGAII and pMSMODE. Moreover, the performance of MOEAs in metric is a little worse than that of pMSMOEAs.
On DTLZ3 function, it could be observed from Table 6 that performance of pMSMOEAs algorithms in both metric and metric has considerable competitiveness over this problem. From Figure 9, it can be seen that the fronts obtained from pMSMOEAs and MOEAs are found to have a basically outstanding result in terms of convergence, while they do not perform satisfactorily in terms of diversity. However, pMSMOEAs are better than MOEAs for the problem.
4.3.3. Time Complexity Analysis
In order to demonstrate the difference in the time complexity of six algorithms, Figure 10 plots the average CPU time over 10 runs.
Figure 10 shows the CPU time for six algorithms and gives the results on six benchmark functions (SCH2, ZDT3, ZDT4, ZDT6, DTLZ2, and DTLZ3). The results show that the time complexity of the original three MOEAs is basically the same, and NSGAII and MOEA/D are slightly better than MODE. The time complexity of pMSMOEAs is slightly higher than that of MOEAs, but within acceptable limits. In addition, for twoobjective functions (SCH2, ZDT3, ZDT4, and ZDT6), the time complexity of pMSMOEAs is about 1.321.64 times that of the original MOEAs. However, as the number of objectives increases, this difference is more pronounced. For example, for threeobjective functions (DTLZ2 and DTLZ3), the time complexity of pMSMOEAs is about 1.381.92 times that of MOEAs. It is worth mentioning that although pMSMOEAs consume more CPU time, there is a significant improvement in performance compared to MOEAs.
5. Application for MultiObjective Portfolio Management
5.1. Introduction of MOPOP
MOPOP has always had an indispensable place in modern risk management. Its ultimate goal is to find an optimal way of distributing a set of available assets, and the budget is scheduled. This is why MOPOP is favored by investors in terms of determining portfolio strategies. However, there is not a single portfolio that could fully satisfy the needs of all investors. In fact, investors' preference for riskreturn ultimately determines a portfolio is an optimal one or not.
5.1.1. TwoObjective Portfolio Model
Markowitz [16] proposed a formal twoobjective portfolio model, which is MV portfolio model. He used the mean returns of assets and covariance of them to describe the investment return and risk, respectively. His model considers two conflicting aspects: maximizing expected return of a portfolio while minimizing its risk [28]. In the existence of risk and return, the optimal solution is not a value yet a set of optimal portfolios, which weighs between above two aspects.
The MV model considered in this paper could be shown as follows [16]:where is variance of portfolios and w_{i} and w_{j} are the weights of assets i and j in all assets, respectively. σ_{ij} is covariance between above two assets. N denotes quantity of available assets, and r expresses expected return while r_{i} represents expected return of the asset i.
Although being popular in the past, the above MV model has an assumption that the return of assets is normally distributed. Unfortunately, the conditions are a bit harsh in real life and are rarely satisfied. There is now a widespread recognition of the fact that portfolios in reality do not follow a multivariate normal distribution. Skewness started to be considered in POP [29]. This implies the risk criteria of MV model could replace the variance with semivariance. Markowitz indeed recommended that models considering semivariance are preferable [30].
5.1.2. ThreeObjective Portfolio Model
In this section, a threeobjective portfolio model, returnriskcost model, is proposed. The two main innovations are the addition of transaction costs and replacement of the risk criterion. There exist three criteria in the proposed model: expected return (measured by mean of assets) that should be maximized, risk (semivariance of assets) that should be minimized, and expected cost (Euclidean distance of weight vectors) that should be minimized.
The MV model has a factbased function limitation; that is, the weight of each asset in the portfolio should be a nonnegative real number and the sum of them should be 1. In the modified Markowitz model by Fernandez and Gomez, the upper and lower limit constraints are added. The constraints of the modified model could be shown as follows:
The semivariance is described in [29] as follows:where B indicates the return after comparison; R_{it} expresses the return of asset i in period t.
An indirect method is employed in this paper to represent transaction costs. There are two assumptions that the transaction cost is related to the quantity of an asset and the distance between current portfolio as well as the expected portfolio is considered in transaction cost. According to the above assumptions, the distance which is the number of different weights in two different portfolios could be quantified as the Euclidean distance:with higher values meaning higher transaction costs.
Hence, the return–riskcost portfolio model proposed in this paper could be formalized as follows:
Among them, budget constraints are given as follows:
It is worth mentioning that the number of model objectives can also be increased if there are suitable indicators.
5.2. Applications for Portfolio Problem
A numerical example is provided to demonstrate the effectiveness of pMSMOEAs for solving MOPOP in this part. A Cesarone’s study shows that limiting the size of portfolio is capable of improving performance [31]. So historical daily data of 12 assets from Shanghai Stock Exchange are employed in the experiment, which are collected by every stock's month rates from January 2010 to December 2016.
Running pMSMOEAs in MATLAB software, under the mentioned above models, the efficient frontiers of the pMSMOEAs are calculated, as shown in Figures 11 and 12. In addition, since the true PF of the portfolio problems is not clear, all the PF of pMSMOEAs algorithms are integrated to obtain a PF as the reference PF (true PF) [32]. The hypervolume indicator has been utilized to evaluate the performance of pMSMOEAs. The results of comparison are listed in Tables 7 and 8.
(a) PF obtained by pMSNSGAII on MV portfolio model
(b) PF obtained by pMSMODE on MV portfolio model
(c) PF obtained by pMSMOEA/D on MV portfolio model
(a) PF obtained by pMSNSGAII on returnriskcost portfolio model
(b) PF obtained by pMSMODEI on returnriskcost portfolio model
(c) PF obtained by pMSMOEA/D on returnriskcost portfolio model
Figure 11 gives the generated nondominated solutions of MV model using pMSMOEAs. Risk and expected return are almost positively correlated, which means the greater risk the investors can accept, the greater expected return they can obtain. Investors could select a portfolio approach based on their preference for risk to get a corresponding return.
Obviously, in Figure 11, efficient curves of pMSNSGAII and pMSMOEA/D are smooth and continuous, while pMSMODE's is converse. From Table 7, it shows that performance of pMSNSGAII, pMSMOEA/D, and pMSMODE in hypervolume indicator is decreasing in order.
Figure 12 gives the generated nondominated solutions of returnriskcost model using pMSMOEAs. The distributions of solutions are located in threedimensional coordinate graphs corresponding to return, risk, and cost. As can be seen, risk and expected return are also almost positively correlated when cost is not considered. Similarly, this is also the case with cost and risk, cost and expected benefit without considering another variable, which is consistent with the reality. When considering these three variables, investors could select a portfolio approach based on their preference for risk and cost to receive corresponding benefits.
Apparently, in Figure 12, the distribution of nondominated solutions in Figure 12(a) as well as Figure 12(c), which resolved by pMSNSGAII and pMSMODE, is more uniform and more diverse in the whole searching space. From Table 8, it can be perceived that the performance of pMSNSGAII is better than pMSMODE and pMSMOEA/D in terms of hypervolume indicator.
6. Conclusion
In this paper, a multiswarm multiobjective optimizer based on poptimality criteria called pMSMOEAs is proposed. In pMSMOEAs, poptimality criteria are employed to ensure algorithms converge to true PF. In addition, multiple swarm cooperative coevolution is adopted, guaranteeing the diversity of the whole population. PMSMOEAs are simply constructed and easily achieved and have considerable potential to solve complex MOP. With six mathematical benchmark functions, pMSMOEAs are proven to obtain good distributed PF with respect to optimization accuracy and convergence robust.
Additionally, a constrained MOPOP with three criteria, expected return (mean of return), risk (semivariance of return), and expected cost (the Euclidean distance of weight vectors), is proposed. And pMSMOEAs are utilized to resolve the returnriskcost portfolio model. The proposed pMSMOEAs are considered to be capable of getting a good diversity of solutions' distribution.
Appendix
SCH2: this is a singlevariable problem having a convex Pareto optimal set. The functions used are as follows:where the variable lies in the range .
ZDT3: this is a 30variable problem having a number of disconnected Pareto optimal fronts:where all variables lie in the range . The Pareto optimal region corresponds to for , and hence not all points satisfying lie on the Pareto optimal front.
ZDT4: this is a 10variable problem having a convex Pareto optimal set.where .
ZDT6: this is a 10variable problem having a nonconvex Pareto optimal set. Moreover, the density of solutions a cross the Pareto optimal region is nonuniform and the density towards the Pareto optimal front is also thin:where all the variables lie in the range . The Pareto optimal region corresponds to and for .
DTLZ2: this test problem has a spherical Paretooptimal front:where the Paretooptimal solutions corresponds to and all objective function values must satisfy . As in the previous problem, it is recommended to use . The total number of variables is is suggested.
DTLZ3: this test problem has a spherical Paretooptimal front:where , and the total number of variables is suggested.
Data Availability
All data in this article is unprocessed raw data and reliable.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
Yabao Hu is the main writer of this paper. She proposed the main idea, proposed pMSMOEAs, and completed the simulation. Hanning Chen gave some important suggestions for the simulation. Maowei He and Liling Sun analyzed the result of the experiments. Rui Liu did some work in the revision of the paper in late period. Hai Shen put forward some constructive opinions on the increase of CPU time information and the revision of the paper and guided the added experiment. All authors read and approved the final manuscript.
Acknowledgments
This work is supported by National key Research and Development Program of China under Grants nos. 2017YFB1103603 and 2017YFB1103003, National Natural Science Foundation of China under Grants nos. 61602343, 51607122, 61772365, 41772123, 61802280, 61806143, and 61502318, Tianjin Province Science and Technology Projects under Grants nos. 17JCYBJC15100 and 17JCQNJC04500, and Basic Scientific Research Business Funded Projects of Tianjin (2017KJ093, 2017KJ094).