Many-Objective Particle Swarm Optimization Using Two ... - IEEE Xplore

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IEEE TRANSACTIONS ON CYBERNETICS, VOL. 47, NO. 6, JUNE 2017

Many-Objective Particle Swarm Optimization Using Two-Stage Strategy and Parallel Cell Coordinate System Wang Hu, Member, IEEE, Gary G. Yen, Fellow, IEEE, and Guangchun Luo

Abstract—It is a daunting challenge to balance the convergence and diversity of an approximate Pareto front in a many-objective optimization evolutionary algorithm. A novel algorithm, named many-objective particle swarm optimization with the two-stage strategy and parallel cell coordinate system (PCCS), is proposed in this paper to improve the comprehensive performance in terms of the convergence and diversity. In the proposed two-stage strategy, the convergence and diversity are separately emphasized at different stages by a single-objective optimizer and a manyobjective optimizer, respectively. A PCCS is exploited to manage the diversity, such as maintaining a diverse archive, identifying the dominance resistant solutions, and selecting the diversified solutions. In addition, a leader group is used for selecting the global best solutions to balance the exploitation and exploration of a population. The experimental results illustrate that the proposed algorithm outperforms six chosen state-of-the-art designs in terms of the inverted generational distance and hypervolume over the DTLZ test suite. Index Terms—Many-objective optimization particle swarm optimization (MaOPSO), many-objective optimization problem (MaOP), parallel cell coordinate system (PCCS), particle swarm optimization (PSO).

I. I NTRODUCTION T IS well documented that most highly regarded multiobjective optimization evolutionary algorithms (MOEAs), such as the nondominated sorting generic algorithm II (NSGA-II) [1] and the advanced strength pareto evolutionary algorithm (SPEA2) [2], perform well in multiobjective optimization problems (MOPs) with two or three objectives. However, they often fail in an optimization problem with a large number of objectives [3], [4]. An MOP with more than three objectives is commonly referred to as a manyobjective optimization problem (MaOP) in the evolutionary

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Manuscript received July 11, 2015; revised November 4, 2015; accepted March 21, 2016. Date of publication April 15, 2016; date of current version May 15, 2017. This work was supported in part by the Fundamental Research Funds for the Central Universities under Grant ZYGX2013J078, in part by the Sichuan Province Science and Technology Support Project under Grant 2015FZ0043, and in part by the China Scholarship Council. This paper was recommended by Associate Editor Y. Tan. W. Hu and G. Luo are with the School of Computer Science and Engineering, University of Electronic Science and Technology of China, Chengdu 611731, China (e-mail: [email protected]). G. G. Yen is with the School of Electrical and Computer Engineering, Oklahoma State University, Stillwater, OK 74075 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TCYB.2016.2548239

multiobjective optimization community. The main reason why these MOEAs lose their search capability in an MaOP is mainly due to the Pareto optimality used to compare the many-objective vectors. The proportion of nondominated solutions based on Pareto optimality among all the solutions found by an evolutionary algorithm (EA) dramatically grows as the number of objectives increases. Thus, the search ability of Pareto dominance-based algorithms is rapidly deteriorated due to the indiscrimination among the found solutions. In addition, it is difficult to represent and visualize the complete Pareto front of an MaOP, because a high-dimensional front requires an exponential number of nondominated solutions with respect to the number of objectives. Over the past few years, some appreciable efforts have been dedicated to tackle these challenges in the domain of MaOP [5]–[26]. Many improvements from the existing MOEAs, such as dimensionality reduction techniques [5], [6], variants of the Pareto-dominance [7]–[10], ranking dominance-based designs [11], [12], indicatorbased approaches [13], [14], and aggregation-based methods [15], [16], are proposed for optimizing the MaOPs. Additional recent works were dedicated to the visualization [17] and performance evaluation [18] of the MaOPs. These contributions provided a promising prospect for solving the MaOPs. Nevertheless, the performance in terms of convergence and diversity of a many-objective optimization EA (MaOEA) is very difficult to reconcile, and it is still far from meeting the requirements of MaOPs. The complexity in balancing convergence and diversity in an MaOEA is much greater than that in an MOEA. Another potential issue in an MaOEA is due to the dominance resistant solutions (DRSs) or outliers [19], [20]. The DRSs are referred to as those nondominated solutions with a poor value in at least one of the objectives but with nearly optimal values in the others. The number of DRSs grows as the number of objectives increases. A DRS can promote the diversity of an approximate Pareto front but incur the dominance resistance to hinder the convergence toward the true Pareto front [3], [20]. It is interesting to note that some single-objective optimization methods are recalled for solving MaOPs or MOPs (e.g., MSOPS [15] and MOEA/D [16]) to emphasize the convergence of an approximate Pareto front. Yet, these aggregationbased MOEAs are subjected to the predefined weight vectors for the diversity of an approximate Pareto front.

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HU et al.: MANY-OBJECTIVE PSO USING TWO-STAGE STRATEGY AND PCCS

In MOEAs, a few works introduced the idea of multiple stages to optimize MOPs [27], [28]. Chang and Chen [28] extended the subpopulation genetic algorithm (SPGA) with a global Pareto archive technique and a two-stage approach into SPGA II to solve MOPs. Inspired by these aggregation-based MOEAs and the idea of devoting multiple stages in balance of exploration and exploitation, a two-stage strategy is formulated for MaOEAs to improve the comprehensive performance in terms of the convergence and diversity. In this strategy, several extreme solutions (ESs) nearby an approximate Pareto front of an MaOP are identified at stage I from the aggregation-based single-objective optimization problems (SOPs). The approximate Pareto front of the MaOP is then extended at stage II from those ESs obtained at stage I by an MaOEA. In order to improve the diversity of the approximate Pareto front, the parallel cell coordinate system (PCCS) [29], [30] is adopted and amended at stage II to manage the diversity of an MaOEA, such as maintaining a diverse archive, identifying the DRSs, and selecting the sparse solutions as the diversity guiders. The remainder of this paper is organized as follows. The related works and the motivations are presented in Section II. A general two-stage framework is outlined for the MaOEAs in Section III. The proposed algorithm, MaOPSO/2s-pccs, is detailed in Section IV. The experimental results are discussed in Section V. And the conclusions are summarized in the last section. II. R ELATED W ORKS AND M OTIVATIONS The aggregation-based MaOEAs and the diversity evaluation mechanisms are briefly reviewed as the background knowledge of this paper. The inspirations of the proposed algorithm are also presented at the end of each related topic. A. Aggregation-Based MaOEAs The simplest aggregation method for an MaOP is the singleobjective optimizers, which multiply the objective values by a weight vector and then accumulate them to a scalar value. The repeated single objective (RSO) with a weighted min-max strategy was proposed in [31] for comparing the performance of MaOEAs. The RSO did not succeed in reaching the true Pareto front due to the small number of function evaluations (only 300) per run and the loss of information with every restart. However, the RSO outperforms the NSGA-II and the SPEA2 in cases of five- and six-objective MaOPs [31]. The MSOPS [15] uses an aggregation method and a ranking scheme based on multiple weight vectors. The weighted min-max strategy was introduced in the MSOPS [15]. And the vector-angle-distance-scaling strategy was proposed in the MSOPS-II [32]. Obviously, the uniformity of the approximate Pareto front obtained by the MSOPS or the MSOPS-II is determined by the predefined weight vectors. The experimental results showed that the MSOPS performed well with respect to the convergence criterion [15]. The MOEA/D [16] is one of the most popular MOEAs nowadays. An MOP or MaOP can be decomposed into a number of subproblems by a group of predefined weight

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vectors. The MOEA/D simultaneously optimizes a number of single-objective subproblems which are defined by the achievement scalarizing function, such as the weighted sum and the weighted Tchebycheff function. The MOEA/D has been demonstrated its advantage in both MOPs [33] and MaOPs [34] when comparing with the NSGA-II. However, as reported in [9], the MOEA/D was outperformed by the MSOPS [15], the ε-dominance-based multiobjective optimization EA (ε-MOEA) [35], and the gridbased EA (GrEA) [9] in most of the benchmark MaOPs in terms of the inverted generational distance (IGD). Although a set of well-distributed weight vectors can be specified to the target directions in the MOEA/D, the corresponding objective solutions cannot be assured to uniformly locate in the approximate Pareto front, especially for those problems with an irregular (i.e., nonuniform) Pareto fronts. From the aggregation-based MaOEAs mentioned above, these algorithms are subjected to the predefined weight vectors for the purpose of a well-distributed Pareto front. However, an important merit of the aggregation-based MaOEAs is to obtain a number of well-converged solutions (not necessary the whole approximate Pareto front) of an MaOP. Inspired from this idea, a few solutions, which approximate the true Pareto front of an MaOP as close as possible, can be identified in advance by an aggregation-based single-objective optimizer in one run or a few separate runs. Then, the more solutions with a good uniformity can be extended from those obtained solutions by a many-objective optimizer. Thus, the convergence and diversity of an approximate Pareto front can be emphasized in different stages in an MaOEA. Of course, a cooperative mechanism for convergence and diversity should be required in different stages. With this motivation, a two-stage framework, to be presented in Section III, is proposed for the MaOPs to improve the convergence and diversity of an approximate Pareto front. B. Diversity Evaluation Mechanisms A diversity evaluation mechanism is an important component in most of MOEAs and MaOEAs because the environmental selection and the external archive maintenance are often depended on it. The crowding distance [1] is a density estimation approach based on the nearest neighbors. However, the crowding distance may lead to the premature convergence, because those solutions close to the ESs in objective space are often preferred to be retained in an external archive [36]. The adaptive grid was proposed in [37] to maintain the diversity of an archive. Yet, the adaptive grid method biases the under-represented areas of an approximate Pareto front. The ε-dominance was proposed as a grid-based technique [38]. The objective space is divided into some hyper-boxes by a user-defined relaxed vector ε, and each hyper-box holds at most one individual. The concept was evolved as a steady state ε-MOEA [35]. The adaptive ε-dominance [39] was further developed in the ε-MOEA to address the issue about the loss of the boundary solutions.

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The GrEA [9] was proposed to strengthen the selection pressure toward the optimal direction, as well as to maintain a well-distributed archive. Those grid-based approaches are the popular techniques for the archive maintenance and environmental selection in terms of convergence, diversity, or both of them. However, in a highdimensional objective space, the KM hyper-boxes are required in these grid-based approaches to accommodate the candidate solutions. The costs of storage and computation for so many hyper-boxes are too expensive to perform the tolerable iterations in an MaOEA. Meanwhile, K in a grid and ε in the ε-dominance are the problem-dependent parameters. It is difficult to specify these parameters by a decision-maker without a priori knowledge of an MaOP. C. Multiobjective Optimization Particle Swarm Optimization Particle swarm optimization (PSO), meta-heuristically inspired by the social behavior of bird flocking or fish schooling, is a population-based stochastic optimization technique developed by Kennedy and Eberhart [40]. PSO is characterized as simple in concept, easy to implement, and computationally efficient when compared with the other heuristic techniques such as genetic algorithm. Naturally, as a singleobjective optimizer, the relative simplicity and the successful applications motivate the researchers to extend its potential ability from SOPs to MOPs [36], [45], [46]. However, apart from the common issue, known as archive maintenance, in MOEAs on retaining the diverse and welldistributed nondominated solutions, there are two particular issues to be addressed when applying PSO to MOPs. The first particular issue in multiobjective PSO (MOPSO) is the update of global best (gBest) and personal best (pBest), because there is not an absolute best solution, but rather a set of nondominated solutions. The second particular issue is the fast convergence, well-known as one of the most important features of PSO, at the early stage of the evolutionary process. However, this positive feature usually is at the cost of a rapid loss of diversity, which leads to the premature convergence or local optima not only in single-objective optimization PSO (SOPSO) but even more seriously, in MOPSO. In [29] and [30], a PCCS was designed to map the nondominated solutions in an archive into a 2-D grid. A nondominated solution can be presented in the PCCS by its parallel cell coordinates (PCCs). The density of a nondominated solution in an archive can be evaluated in the PCCS for maintaining the archive and selecting the global best solutions (gBests). In this paper, the PCCS will be adopted again and amended further to solve the MaOPs, such as maintaining a diverse archive, identifying the DRSs, and selecting the gBests. The main contributions of this paper are as follows: 1) a two-stage framework for an MaOEA is proposed to emphasize the convergence and diversity at separate stages during the evolutionary process; 2) a many-objective optimizer based on PSO is designed for tackling the MaOPs; 3) the PCCS is enhanced to evaluate the diversity for maintaining the archive, identifying the DRSs, and selecting the diverse-gBests (d-gBests); and 4) a leader group, comprising

Fig. 1.

Two-stage framework proposed for an MaOEA.

of convergence-gBests (c-gBests) and d-gBests, is designed for selecting the gBest for a PSO population to balance exploitation and exploration. III. F RAMEWORK FOR T WO -S TAGE M AOEA As analyzed in Section II-A, the aggregation-based MaOEAs are effective at the convergence. With this inspiration, a two-stage optimization strategy is proposed for the MaOPs to emphasize the convergence and diversity of an MaOEA in two different stages. The two-stage framework is illustrated in Fig. 1. At stage I, the M (the number of objectives) SOPs or subproblems are firstly aggregated from an MaOP. Then, the optimal solutions of these SOPs, namely the ESs near to an approximate Pareto front, can be sought out by a single-objective optimizer at M separate runs. At stage II, the population of a manyobjective optimizer is randomly initialized around those ESs. The initialized population is then guided by the ESs while the evolutionary process is iterating. At the same time, the ESs will be updated if a new solution is better according to a certain criterion, such as the variants of Pareto dominance. A. Stage I: Generating the Extreme Solutions The goal of stage I is to generate the M ESs, which are expected to lie as near to the true Pareto front of a given MaOP as possible. Each ES can be found by a single-objective optimizer from an SOP, which is aggregated from an MaOP by M appropriate weights. In this framework, a single-objective optimizer can be one of the nature-inspired meta-heuristic algorithms, such as PSO, GA, DE, and etc. As a case study in this paper, PSO is selected to serve as the single-objective optimizer due to its fast convergence and relative simplicity. It is difficult to choose a set of universal weight vectors for a general MaOP to generate the ESs. Yet, the weight matrix, WM , with M rows for weight vectors and M columns for


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objectives is recommended as ⎡ ⎤ 1−ε 1−ε ··· ⎢ ε M−1 M−1⎥ ⎢ 1−ε 1−ε ⎥ ⎢ ⎥ ε ··· ⎢ ⎥ M−1 M − 1 ⎥. (1) WM = [wij ] = ⎢ ⎢ . .. .. ⎥ .. ⎢ .. ⎥ . . . ⎥ ⎢ ⎣ 1−ε ⎦ 1−ε ··· ε M−1 M−1 In (1), ε approaches to zero in theory. However, to avoid the infinitesimal in programming the algorithm, ε can be a very small positive number so as to find an ES in 1-D, for example, ε = 1e-6. The sum of the weight components wij in each row i is equal to one, namely, M j=1 wij = 1. Thus, the ith SOP can be aggregated from the M objective functions and the weight matrix, WM SOPi (x) =

M

wij × fj (x).

Fig. 2.

Three ESs of DTLZ1 found at stage I.

(2)

j=1

Here, i = 1, 2, . . . , M; wij is the weight at the ith row and the jth column; fj (x) is the jth objective value of an individual x. Certainly, the other aggregation methods, such as Chebyshev or penalty-based boundary intersection [16], can be also used to transform an MaOP into M SOPs to search for the M ESs. To validate the effectiveness of (1), the weight matrix W3 is used to a three-objective problem as a verifiable experiment. After the test instance DTLZ1 [41] with three objectives is aggregated into three SOPs by W3 , the three ESs, (0.5,0,0), (0,0.5,0), and (0,0,0.5) are sought out in this experiment by the SOPSO at three separate runs. These three ESs of DTLZ1, illustrated in Fig. 2, are just located at the three vertices of the true Pareto front of DTLZ1, respectively. It should be noted that not all the ESs can be exactly located at the vertices of a true Pareto front. Yet, the ESs can be expected to be closely located around the true Pareto front if the number of iterations is sufficient. However, the computational cost will grow as the number of iterations increases. So, an improvement mechanism for the ESs is necessary at stage II to balance the accuracy and complexity. B. Stage II: Extending to the Approximate Pareto Front The goal of stage II is to obtain the approximate Pareto front of an MaOP as accurate and uniform as possible by a manyobjective optimizer. The population of a many-objective optimizer based on the two-stage framework is required to be randomly initialized around the ESs obtained at stage I, rather than in the whole decision space of an MaOP. In this paper, the many-objective optimization PSO (MaOPSO) is chosen as the many-objective optimizer through extending our previous works on the MOPs [30] to handle the MaOPs. Although the PSO-based method is selected at both stage I and II here to design a new MaOPSO, the different EAs can be separately chosen to implement the two-stage framework as a hybrid structure. To be sure, the other optimization algorithms, i.e., DE, GA, etc., will be easily used as the single-objective optimizer

in stage I. However, some corresponding modifications are required for them if used in stage II, such as selection and crossover operators in GA. For the consideration of computational cost, the ESs obtained at stage I might not be well converged to the true Pareto front of an MaOP. In this case, the ESs might not have the adequate abilities to guide the whole population to approach the true Pareto front. Therefore, it is necessary for the ESs to be continuously improved at stage II to compensate for the inadequate accuracy of stage I. An ES will be updated by a new solution if the new solution is better than the old one according to a certain criterion. In a PSO, a better solution can be easily found in the regions around the gBests because most particles will be recruited to search those regions intensively. Therefore, the ESs can be used to improve the approximate front if they are selected as the gBests in a PSO-based MaOEA. To illustrate the critical role of stage I in the two-stage framework, a comparative experiment for an MaOEA with and without stage I is presented in Fig. 3. The approximate Pareto fronts, plotted through parallel coordinates, are optimized by the MaOPSO/2s-pccs (to be described in Section IV) from the five-objective DTLZ1. Fig. 3(a) and (b) are respectively obtained by the MaOPSO/2s-pccs without stage I and with stage I under the same number of function evaluations. From Fig. 3, it can be seen that the convergence of Fig. 3(b), guided by the ESs, is much better than that of Fig. 3(a), because the ranges of objective function values in Fig. 3(a) and (b) are [0,230.0] and [0,0.5], respectively. Here, DTLZ1 is a minimization problem. So, the smaller the objective value is, the better the approximate Pareto front is convergent to the true Pareto front. Apparently, the approximate Pareto font in Fig. 3(b) is much better than that in Fig. 3(a). Therefore, stage I is necessary and important for the proposed MaOEA to obtain a well-converged Pareto front. IV. P ROPOSED A LGORITHM : M AOPSO/2 S - PCCS As an implementation of the two-stage framework, the PSO is selected for both the single-objective optimizer at stage I

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Fig. 4.

Architecture of the proposed MaOPSO/2s-pccs.

components in the architecture of the MaOPSO/2s-pccs are respectively described in the following sections. A. SOPSO Fig. 3. Experimental example for illustrating the effectiveness of stage I. (a) Without stage I. (b) With stage I.

and the many-objective optimizer at stage II. In this paper, the proposed algorithm is abbreviated as MaOPSO/2s-pccs to highlight the two-stage framework and the PCCS. The architecture of MaOPSO/2s-pccs is presented in Fig. 4. The two-stage framework is distinctly expressed by a dashed line at the middle of Fig. 4 for separating stages I and II. The M ESs are conveyed from stages I to II for improving the convergence of this algorithm. At stage I, an MaOP is firstly aggregated into M SOPs by the weight matrix WM in (1). The M ESs are then sought out from the M SOPs by the SOPSO over M separate runs. At stage II, a leader group, comprising of the d-gBests and the c-gBests for selecting the gBest, is designed for balancing the exploitation and exploration of a PSO population in the MaOPSO. The ESs from stage I act as the c-gBests at stage II to promote the convergence of the MaOPSO. Also, an ES is continuously improved at stage II if a better solution is found by the MaOPSO. The d-gBests in the leader group are selected from those nondominated solutions with the better diversity in the archive according to the PCCS. If a new solution found by a particle is better than those existing solutions according to a comparison criterion, it will update the personal best solution (pBest) of the particle and the nondominated solutions in the archive. All particles in the MaOPSO are iterated by a flight controller. The main

At stage I, the M SOPs, aggregated from a given MaOP, can be optimized by an SOPSO at M separate runs. Yet, any parallel computational techniques, not to be discussed further here, for optimizing the M SOPs at one run might reduce the runtime of stage I. The pseudocode of the SOPSO is presented in Algorithm 1. From lines 2 to 20, the SOPSO with a restart strategy is repeatedly performed M times to seek out the M ESs. For simplicity, the inertia weight ω can be linearly decreased from 0.9 down to 0.4 [42] with respect to the generation loop variable g. The personal acceleration factor c1 and the social acceleration factor c2 can be set to the constant 1.429 [43]. Furthermore, the elite learn strategy (ELS) [44] is used at line 13 to enhance the ability for escaping from the local optima in the multimodal optimization problems. The learning rate, lr, in ELS can be linearly decreased from 1.0 down to 0.1 with respect to g [44]. In Algorithm 1, the number of function evaluations in SOPSO at stage I is M × GI × NI , where GI is the maximal generations allowed at stage I, and NI is the size of the population at stage I. In order to obtain the satisfied ESs at a rational computational cost, GI is set to the number of generations at stage II, and the population size NI is set to 20, which is widely used in an SOPSO [44]. B. PCCS The PCCS was proposed in [29] and [30] as a diversity evaluation mechanism for maintaining a well-distributed


Algorithm 1: SOPSO in Stage I for Generating the ESs

1

2 3

4 5 6

7 8 9 10

11 12 13 14 15 16

17 18 19 20 21 22

Input: 1) The MaOP with M objective functions fm (m = 1, 2, . . . M) and the search space SD (D is the number of decision variables); 2) The weight matrix WM ; 3) The population size NI and the max generations GI at Stage I. Output: The M ESs. Generate the M SOPm by aggregating fm with WM according to Eq. (2); For m=1 to M Initialize randomly positions Xn =[xn,1 , xn,2 , . . . , xn,D ]T , n=1, 2, . . . , NI , and velocities vn =[vn,1 ,vn,2 ,. . . ,vn,D ]T of the population in SD ; Evaluate the initial fitness, fSOPm (xn ), of each particle; Initialize personal best, pBestn , for the n-th particle by xn ; Initialize global best, gBest, by the particle with the optimal fitness; For g=1 to GI Adjust ω according to the linear decreasing inertia weight; For n= 1 to NI vn = ω×vn + c1 ×r1 ×(pBestn −xn ) + c2 ×r2 ×(gBest−xn ); xn =xn + vn ; If rand()< lr /* lr is the learning rate */ An elite learn strategy (ELS) is applied to xn ; End; Evaluate the new fitness, fSOPm (xn ), for the n-th particle; Update pBestn if fSOPm (xn ) is better than the fitness of pBestn ; Update gBest if fSOPm (xn ) is better than the fitness of gBest; End; End; Set gBest to ESm ; End; Return the M ESs.

archive and selecting the gBest for a particle. In this paper, the PCCS is significantly enhanced to challenge the MaOPs. For convenience and completeness, the concepts in PCCS are briefly described here. The details of PCCS can be referred in [29] and [30]. In the PCCS, the mth objective of the kth nondominated solution in an archive, fk,m , can be mapped to an integral label number within a 2-D grid with K×M cells according to (3), where K is the current size of the archive

fk,m − f min (3) Lk,m = K max mmin . f m − fm Here, x is a ceiling operator that returns the smallest integer, which is not less than x. fmmax = maxk fk,m and fmmin = mink fk,m are the maximum and minimum of the mth objective value in the archive, respectively. Lk,m ∈ {1, 2, . . . , K} is an integer number transformed from the real number, fk,m . The distance between two solutions in the PCCS, named parallel cell distance (PCD), is measured by the sum of the differences of cell coordinates over all objectives. The PCD of two nondominated solutions Pi and Pj , PCD(Pi ,Pj ), can be calculated according to

0.5 if ∀m Li,m = Lj,m (4) PCD Pi , Pj = M L − L j,m otherwise. m=1 i,m

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The density of Pi , in the objective hyperspace, can be measured by the PCD between Pi and all other members, Pj ( j = 1, 2, . . . , K, j=i), in the archive according to Density(Pi ) =

K j=1 j=i

1

2 . PCD Pi , Pj

(5)

The potential, to be used in Section IV-C, of a nondominated solution Pi in an archive is defined by Potential(Pi ) =

M

Li,m .

(6)

m=1

C. Archive An external elitist archive is an important feature in most of the current MOEAs and MaOEAs. An appropriate archiving strategy is required for maintaining an accurate and well-distributed approximate Pareto front. As mentioned in Section I, the number of DRSs in an MaOEA grows quickly as the number of objectives increases. The DRSs can promote the diversity of an approximate Pareto front. However, they are harmful for an MaOEA because the DRSs can incur the dominance resistance [3], [20] to hinder the convergence toward the true Pareto front. Therefore, it is necessary to identify and eliminate the DRSs when maintaining an archive. Suppose that the difference of potential at the mth objective can be defined for a nondominated solution Pi as K j=1 Lj,m (7) Li,m = Li,m − . K According to the concept of DRS defined in Section I, a DRS can be discriminated from a set of nondominated solutions by its potential, defined in (6). For a minimizing MaOP, the PCCs of a DRS in the minority of the objectives (at least one objective, say f 1 ) are much greater than those of the other nondominated solutions (namely non-DRSs) according to (3). In addition, the PCCs of this DRS in the other objectives (say f2−M ) are much smaller than those of the non-DRSs. So, for this DRS, the sum of differences of potential in all objectives, Potential(DRS), will be much greater than that of the non-DRSs according to Potential(DRS) = L1 +

M

Lm .

(8)

m=2

Here, L1 and Lm are the differences of potential of the DRS at the first and mth objective, respectively. Thus, the Potential can be used to identify a DRS from an archive. If a nondominated solution Pi in an archive satisfies the condition1 in (9), Pi can be identified as a DRS and be eliminated 1 The Potential for a nondominated solution P can be calculated by i Potential(Pj ))/K)| if (6) and (7) Potential(Pi ) = |Potential(Pi ) − (( K j=1 M M are substituted in Potential(Pi ) = m=1 Li,m = | m=1 Li,m − K M (( j=1 ( m=1 Lj,m ))/K)|.

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D. Leader Group

Algorithm 2: Archive Maintenance Algorithm Input: 1) An archive A with a bounded size L to be maintained; 2) The ESs in a guider set G; 3) A new solution p. Output: The updated archive A. 1. If A=Ø then A={p}; Goto 12; 2. If p is dominated by any a∈A then goto 12; 3. For each a∈A 4. If a is dominated by p, then A=A/{a}; 5. If |A| μDensity (A) + σDensity (A).

(10)

Here, μDensity (A) and σDensity (A) are the mean value and the standard deviation of density of all the nondominated solutions in an archive A. The number of the satisfied solutions to be selected as the d-gBests is about (1−P(μ−σ < x ≤ μ+σ ))× |A|, where P(μ − σ < x ≤ μ + σ ) ≈ 68.27% is the area ratio of the interval [μ − σ, μ + σ ] in a normal distribution with an average μ and a variance σ in the theoretical probability. Thus, there are M c-gBests (namely ESs) and 31.73% × |A| d-gBests in the leader group as the candidates of gBest for the whole population. Here, x is a floor operator that returns the largest integer which does not exceed x. For a specific particle in a generation, the gBest is determined from the leader group by two steps. At first, the type of c-gBest or dgBest is randomly selected at the identical probability, 50%, for balancing the convergence and the diversity of a population. Then, a member is randomly selected from the chosen set of c-gBest or d-gBest for a specific particle at one generation. An example for the leader group is illustrated in Fig. 5. For simplicity, there are only 15 nondominated solutions marked


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Algorithm 3: MaOPSO at Stage II for Obtaining an Approximate Pareto Front

Fig. 5. Example for the leader group. The c-gBests guide some particles to exploit the regions around the ESs so as to search for the better ESs toward the true Pareto front. Meanwhile, the d-gBests guide some other particles to explore the sparser regions so as to extend the approximate Pareto front to the whole objective space.

by squares in the approximate Pareto front. The three c-gBests (marked by circles) are those ESs, sought out by the SOPSO at stage I and shown in Fig. 2. The three d-gBests (marked by triangles) are selected according to (10) from the approximate Pareto front excluding those three ESs. From Fig. 5, the c-gBests guide some particles to exploit the regions around the ESs so as to seek out the better ESs toward the true Pareto front. At the same time, the d-gBests guide some other particles to explore the sparser regions so as to extend the approximate Pareto front to the whole objective space. It should be noted that the archive maintenance strategy in Algorithm 1 will improve the uniformity of the approximate Pareto front according to the density in the PCCS.

E. Personal Best Similar to the gBest, a set of personal nondominated solutions can act as the candidates of the pBest. A personal archive to store the pBest nondominated solutions will be better than only one nondominated solution for selecting the pBest [45]. However, it is too time-consuming to maintain the N personal archives in an MaOPSO. So, in this paper, only one personal nondominated solution is retained in the MaOPSO/2s-pccs as the pBest for a particle. The pBest strategy with respect to the unique personal nondominated solution is widely adopted in MOPSOs [36], [46]. The pBest will be replaced by a new solution if the current pBest is dominated by the new one, or they are nondominated with respect to each other. In the latter case, the newly found solution is more suitable to be selected as the pBest, because it stands for the new situation of the population.

Input: 1) The MaOP with M objective functions fm (m=1,2,. . . ,M); 2) The population size NII and the maximal generation GII at Stage II; 3) The search space SD (D is the number of decision variables); 4) The bounded size L of an archive; 5) The ESs in a guider set E. Output: An approximate Pareto front stored in an archive A. /* Initialize the population */ 1. Initialize xn of the NII particles around ESs obtained by Algorithm 1; 2. Evaluate the initial objective values fn =[f 1 (xn ),f 2 (xn ), . . . ,fm (xn )]T ; 3. Initialize pBestn by xn and fn of the n-th particle; 4. Initialize an archive A by all the nondominated fn for the population; /* Iterate the population*/ 5. For g = 1 to GII 6. Set c_gBest =G and d_gBest={a |a∈A and a satisfies Eq. (10)}; 7. Adjust ω according to the linear decreasing inertia weight; 8. For n = 1 to NII 9. Select a gBest for the n-th particle from the leader group; 10. vn = ω×vn +c1 ×r1 ×(pBestn −xn )+c2 ×r2 ×(gBest−xn ); 11. xn = xn + vn ; 12. An elite learn strategy (ELS) is applied to xn ; 13. Evaluate the new objective values fn =[f 1 (xn ), f 2 (xn ), . . . , fm (xn )]T ; 14. Replace pBestn by the new solution xn if the former is dominated by the latter or they are nondominated w.r.t. each other; 15. Update A by Algorithm 2; 16. Replace the ES e∈E with the new solution s if s is better than e; 17. Return A.

F. Flight Controller After the gBest is selected from the leader group and the pBest is updated in the last generation, the velocity and position of a particle can be updated by the flight equations of PSO. The ELS [44] is also used at stage II to improve the search ability when dealing with a multimodal MaOP. The flight equation of PSO with the ELS is called the flight controller in Fig. 4. As at stage I, the inertia weight is self-adjusted according to the linear decreasing strategy [42], and the personal acceleration factor c1 and the social acceleration factor c2 are also set to a constant 1.429 [43] at stage II in this algorithm for simplicity. G. MaOPSO In the framework of two-stage MaOEA, a many-objective optimizer is required to generate the approximate Pareto front of an MaOP. The PSO-based meta-heuristic algorithm is also selected at stage II as the many-objective optimizer, explained in Section III-B. Based on the ES, the PCCS, and the leader group, a novel MaOPSO is designed in Algorithm 3 to act as the manyobjective optimizer at stage II to search for the well-distributed approximate Pareto front of an MaOP. At line 1 in Algorithm 3, an MaOPSO population is initialized around the ESs. The candidates of c_gBests and d_gBests

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are selected at line 6 according to the leader group strategy presented in Section IV-D. The gBest for a particle at a generation is randomly selected from the candidates at line 9. In this algorithm, the ELS is also applied at line 12 to disturb the particle for escaping from a local optimal. The archive maintenance strategy in Algorithm 3 is called at line 15 to make those overcrowded regions sparse. At line 16, an ES is improved at every generation if there is a better solution found by the population. H. Complexity Analysis The proposed algorithm, MaOPSO/2s-pccs, comprises of the SOPSO and the MaOPSO. The complexity of SOPSO in Algorithm 1 is O(M×GI ×NI ), analyzed in Section IV-A. The complexity of MaOPSO in Algorithm 3 is O(M×L2 ×GII ×NII ), because the archive maintenance algorithm with the complexity of O(ML2 ) in Section IV-C. Therefore, the total complexity of the MaOPSO/2s-pccs is O(M×(GI ×NI +L2 ×GII ×NII )). Here, GI and GII are the maximal generations allowed at stages I and II, respectively; NI and NII are the population sizes at stages I and II, respectively. V. E XPERIMENTS A. Experimental Design 1) Test Problems: The DTLZ test suite [41] including seven scalable test functions DTLZ1-7 is a widely used benchmark for testing MaOEAs. Some complicated characteristics of true Pareto fronts, such as nonconvexity, multimodality, disconnection, and nonuniformity, are covered in the test instances to examine the optimization ability of an algorithm. In this experiment, the test instance of DTLZi (i = 1, 2, . . . , 7) with M objectives is represented in the form of DTLZi(M), where M is set to be 3, 4, 5, 7, and 10. The DTLZi(3) and DTLZi(4) are used to inspect the compatible ability of an MaOEA in a low-dimensional objective space. The dimension of decision space of DTLZi(M) is set to M-1+k, where k is set to be 5, 20, and 10 for DTLZ1, DTLZ7, and DTLZ2-6, respectively, according to the suggestions in [41]. 2) Peer Algorithms: In order to validate the performance of the proposed MaOPSO/2s-pccs, the six state-ofthe-art MaOEAs (i.e., MSOPS [15], IBEA [13], HypE [47], ε-MOEA [35], MOEA/D [16], and NSGA-III [22]) are chosen as the peer competitors. The MSOPS and MOEA/D are the aggregation-based MaOEAs, while the HypE and IBEA are the indicator-based MaOEAs. The ε-MOEA is based on the ε-dominance, a variant of Pareto dominance. The NSGA-III is a newly published MaOEA based on reference points and the nondominated sorting. 3) Performance Metrics: In this experiment, the two performance metrics, IGD [16], and hypervolume (HV) [48], [49], are chosen to quantify the performance in terms of convergence and diversity of an approximate Pareto front. For the IGD metric, the true Pareto fronts of the test instances are required as the referenced fronts to measure the performance metric. The more the samples of a true Pareto front are available, the better the IGD metric will be for an MaOP, yet the

higher the computational cost will be incurred. For balancing the accuracy and complexity, the size of samples in a true Pareto front is set to around (M-2)×5000, namely, about 5000, 10000, 15000, 25000, and 40000 samples for the three-, four-, five-, seven-, and ten-objective problems, respectively. For the HV metric, the relative HV with an acceptable tolerance is approximately calculated through the Monte Carlo simulation with 100 000 sampling points [49]. Each objective value of the reference points for evaluating HV is chosen as the integer value 11 and 22 for the test instances with 3 and 4–10 objectives, respectively. 4) Simulation Settings: The simulation settings of these peer algorithms are set as follows. The size of population is set to be 100 and the maximal generation is set to be 400, so that the total number of function evaluations is 40 000 for all test instances. At stage II of MaOPSO/2s-pccs, the population size is set to be 50 and the maximal generation is set to be 100, so that the total number of function evaluations is 5000. At stage I of MaOPSO/2s-pccs, the population size is set to be 20 and the maximum generation is set to be 400 (the same as the maximum generation in the common simulation settings of the peer algorithms), so that the total number of function evaluations is 8000×M for a test instance with M objectives. Therefore, the sum of function evaluations of MaOPSO/2s-pccs at stages I and II altogether is 8000 × M + 5000. The total number of function evaluations of MaOPSO/2s-pccs will be greater than that of its competitors when the number of objectives is equal to or more than five. From the total of function evaluations, it seems unfair for the competitors of MaOPSO/2s-pccs, especially in those test instances with a larger number of objectives. However, the complexity of SOPSO at stage I is much smaller than that of MaOPSO at stage II according to Section IV-H. The special parameters, such as ω, c1 , and c2 , in the MaOPSO/2s-pccs are explained in Sections IV-A and IV-F. In the ε-MOEA, the ε value is set according to [50], so that the number of nondominated solutions in the archive is close to 100. In the MOEA/D, the parameter T (the number of neighboring subproblems) is set to be 20, and the parameter H (the number of total subproblems) is set to be 13, 6, 5, 3, and 2 for 3-, 4-, 5-, 7-, and 10-objective problems, respectively, so as to make the population size equal to the closest integer 100. The six competed algorithms are GA-based MaOEAs, in which the simulated binary crossover (SBX) and the polynomial mutation are used. The distribution indexes in SBX and the polynomial mutation are both set to be 20 according to the practice in [1] and [22]. To be consistent, the special parameters of these GA-based algorithms are set according to [16], where the MOEA/D worked well on some instances of DTLZ. The crossover rate is set to be one, and the mutation rate is set to be 1/D, where D is the number of decision variables. The weight vectors in the MSOPS and the MOEA/D are uniformly chosen according to their original papers so as to make the obtained approximate Pareto front well-distributed. The performance metrics in terms of IGD and HV on each test instance are obtained from 30 independent runs. In order to provide the statistical quantifications on performance metrics, the nonparametric statistical hypothesis


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TABLE I C OMPARISONS IN T ERMS OF IGD B ETWEEN THE P ROPOSED M AOPSO/2 S - PCCS AND OTHER M AOEA S ON U-T ESTS

test, Mann-Whitney-Wilcoxon rank-sum test [51] (also called U-test), is applied to quantify the superiority of two completing algorithms in a statistical meaningful sense. All simulations in this experiment are performed on a 64-bit notebook PC with 1.2 GHz dual core CPU and 4 GB memory. B. Experimental Results on IGD The experimental results in terms of IGD are listed in Table I. The three items, mean, standard deviation (Std.), and significance symbol of U-test, are filled in the form of “mean (Std.) #” in each data cell of Table I. Here, “#” stands for one of the significance symbol (“+,” “−,” or “=”) of U-test. These symbols respectively implies that the IGD of MaOPSO/2s-pccs is better, worse, or same, than/as that of its competitor according to the U-test at the significance level of α = 5% for a two-tailed test. For example, the experimental data at the first row and the second column in Table I is “6.56E-1 (5.9E-1) +.” Here, the mean and standard deviation of IGD obtained by MSOPS on DTLZ1(3) over 30 runs is 6.56E-1 and 5.9E-1, respectively. The significance symbol is +, which implies that the MaOPSO/2s-pccs performs significantly better than the MSOPS on DTLZ1(3).

The numbers of the same significance symbols (+, −, and =) are respectively summed up from each column and filled in the last three rows with the titles, “Better(+),” “Worse(−),” and “Same(=),” respectively. For IGD, the smaller metric value indicates the better performance of an algorithm. The best value of IGD among all algorithms on each test instance is highlighted by boldface at each data row in Table I. From Table I, the MaOPSO/2s-pccs gains 22 best values of IGD out of 35 test instances, while the MOEA/D, IBEA, MSOPS, NSGA-III, and HypE merely obtain seven, three, two, one, and one best values of IGD, respectively. Yet the ε-MOEA does not produce any best value of IGD. The maximum number of best values gained by the MaOPSO/2s-pccs indicates that the proposed algorithm is much better than its competitors as a whole. According to the U-test, the MaOPSO/2s-pccs performs significantly better (+) than the ε-MOEA, HypE, MSOPS, MOEA/D, NSGA-III, and IBEA, respectively on 32, 30, 27, 27, 25, and 20 test instances over all 35 test instances. Correspondingly, the MaOPSO/2s-pccs is only worse (−) than the ε-MOEA, NSGA-III, HypE, MSOPS, MOEA/D, and IBEA, respectively on 3, 4, 4, 5, 8, and 9 test instances over all 35 test instances. These statistical results from U-test show that the proposed MaOPSO/2s-pccs outperforms its competitors on the DTLZ test suite in terms of

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TABLE II C OMPARISONS IN T ERMS OF HV B ETWEEN THE P ROPOSED M AOPSO/2 S - PCCS AND OTHER M AOEA S ON U-T ESTS

IGD overall. These results justify that the integrated scheme of the two-stage strategy, the PCCS, and the leader group is very effective in improving the comprehensive performance in terms of convergence and diversity. In terms of an individual test function, the MaOPSO/2s-pccs performs best among all the peer algorithms on DTLZ1 (multimodal), DTLZ3 (multimodal), and DTLZ7 (multimodal and disconnected) over all tested numbers of objectives (M ∈ {3, 4, 5, 7, 10}). These results indicate that the MaOPSO/2s-pccs is more effective for the MaOPs with the multimodal or disconnected Pareto fronts. For the multimodal MaOPs, the ESs obtained at stage I of MaOPSO/2s-pccs can improve the convergence of an approximate Pareto front through initializing the population around the ESs at stage II. Moreover, it is easier to obtain a better ES in terms of convergence in a multimodal MaOP by a single-objective optimizer than by a many-objective optimizer. Therefore, stage I is effective for an MaOP in the proposed two-stage framework, especially for a multimodal MaOP. However, the MaOPSO/2s-pccs underperforms its most competitors on DTLZ5-6 (with the degenerated Pareto fronts) with 4-, 5-, 7-, and 10-objectives. The reason for the poor performance of MaOPSO/2s-pccs is that the d-gBests in the leader group

may incidentally attract some individuals to fly away from a degenerated Pareto front to improve the diversity. From the viewpoint of the number of objectives, the MaOPSO/2s-pccs outperforms its competitors on all test functions (DTLZ1-7) with three objectives. It shows that the proposed MaOPSO/2s-pccs works well in the MOPs with low-dimensional objectives. C. Experimental Results on HV The detailed experimental results in terms of HV are presented in Table II. The symbols and the representation style in Table II are the same as those in Table I. For HV, a larger metric value indicates a better performance of an algorithm. From Table II, the proposed MaOPSO/2s-pccs obtains 19 best values of HV out of 35 test instances. The number of best values of HV obtained by the MaOPSO/2s-pccs is slightly fewer than that of IGD, because the subtle differences (with the same mean value and the different Std.) lead to lost the first place on several test instances, such as DTLZ1(4), DTLZ2(3,4), DTLZ4(4,5,7,10), DTLZ5(3), and DTLZ6(3). From the viewpoint regardless of the subtle differences from Std., the proposed MaOPSO/2s-pccs only underperforms


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Fig. 6. Parallel coordinates of the approximate Pareto fronts of DTLZ1(10) obtained by the seven competing algorithms. (a) MaOPSO/2s-pccs. (b) MSOPS. (c) IBEA. (d) HypE. (e) ε-MOEA. (f) MOEA/D. (g) NSGA-III.

the MOEA/D on DTLZ5(4,5,7,10) and DTLZ6(4,5,7,10), the MSOPS on DTLZ5(5,7,10), the IBEA on DTLZ5(4,5,7,10), and the HypE on DTLZ5(10), because the mean values of HV obtained by the MaOPSO/2s-pccs are smaller than those by the latter competitors. In the MaOPSO/2s-pccs, the d-gBests in the leader group cause some individuals fly away from a degenerated Pareto front in DTLZ5-6, which improves the diversity yet weakens the convergence ability. As a whole, the MaOPSO/2s-pccs outperforms the εMOEA, NSGA-III, HypE, MSOPS, IBEA, and MOEA/D, respectively, on 34, 30, 28, 25, 24, and 18 test instances over all 35 test instances in terms of HV. The overall performance of the MaOPSO/2s-pccs on HV is similar to that on IGD. The two

comprehensive metrics of IGD and HV in these experiments indicate that the proposed MaOPSO/2s-pccs is the best one in terms of convergence and diversity among all chosen MaOEAs on the 35 test instances. D. Comparative Approximate Pareto Front To intuitively illustrate the results in terms of convergence and diversity, the parallel coordinates of the approximate Pareto fronts obtained by these peer algorithms on DTLZ1(10) are visualized in Fig. 6. It is noted that the ranges of the vertical axis (objective value) are not the same in all subfigures.

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In Fig. 6, the MSOPS, HypE, and ε-MOEA cannot converge to the true Pareto front of DTLZ1(10), whose maximum at each objective is 0.5. On the other hand, the objective values of the MOEA/D in Fig. 6(f) are much less than 0.5 at each objective dimension implying the approximate Pareto front obtained by the MOEA/D can converge to the true Pareto front, but cannot extend to the full objective space covered by the true Pareto front of DTLZ1(10). The approximate Pareto front obtained by the IBEA in Fig. 6(c) is nearly convergent to one solution in the objective space, as the same observations reported in [51]. This result indicates the diversity of IBEA is the worst one among all peer algorithms. In Fig. 6(a), the maximum of each objective values is almost 0.5. Meanwhile, the connected lines between two adjacent objectives in Fig. 6(a) are more uniform than those of the other subfigures by intuition. So, the approximate Pareto front obtained by the MaOPSO/2s-pccs in Fig. 6(a) is the best one among the six competing algorithms with respect to the tradeoff between convergence and diversity. Visually, from Fig. 6, the diversity of the indicatorbased (IBEA and HypE) and aggregation-based (MSOPS and MOEA/D) MaOEAs are relatively poor, while the convergence of the ε-MOEA is relatively weak. On the other hand, the convergence and diversity of the proposed MaOPSO/2s-pccs can make a delicate balance by combining the proposed comprehensive strategies. In summary, the proposed MaOPSO/2s-pccs is the best design in terms of convergence and diversity measured by IGD and HV among all chosen MaOEAs on the 35 test instances. The experimental results indicate that the convergence and the diversity are well-balanced by stages I and II, respectively. For the peer competitors of MaOPSO/2s-pccs, some algorithms are good at the convergence because of the aggregation-based strategies adopted in the designs such as the MOEA/D and MSOPS, yet perform poorly due to the lack of an effective diversity mechanism. On the other hand, some other peer algorithms emphasize the diversity, such as IBEA, ε-MOEA, and HypE, but provide no compatible design to facilitate the convergence to the true Pareto front in an MaOP. Yet, the proposed MaOPSO/2s-pccs is defeated by most competitors on the test functions with a degenerated Pareto front, such as DTLZ5 and DTLZ6. In this scenario, some particles might be incidentally attracted to fly away from a degenerated Pareto front for improving the diversity but deteriorating the convergence. VI. C ONCLUSION A novel algorithm named MaOPSO/2s-pccs is proposed in this paper for handling the MaOPs to improve the comprehensive performances in terms of convergence and diversity. Inspired from the ideas of aggregation-based and multiplestage-based MOEAs, a two-stage optimization framework is proposed for the MaOPs to emphasize the convergence and diversity in the separate stages. A single-objective optimizer is used in stage I to identify the ESs. A many-objective optimizer is served at stage II to extend the approximate Pareto front from those ESs to the objective space. In order to improve the diversity of an approximate Pareto front, the PCCS is enhanced to evaluate the diversity for maintaining the archive,

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