Population statistics for particle swarm optimization_ ...

Swarm and Evolutionary Computation 22 (2015) 15–29

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Swarm and Evolutionary Computation journal homepage: www.elsevier.com/locate/swevo

Regular Paper

Population statistics for particle swarm optimization: Hybrid methods in noisy optimization problems Juan Rada-Vilela a,b,n, Mark Johnston a,c, Mengjie Zhang a,b a

Evolutionary Computation Research Group, Victoria University of Wellington, PO Box 600, Wellington 6140, New Zealand School of Engineering and Computer Science, Victoria University of Wellington, PO Box 600, Wellington 6140, New Zealand c School of Mathematics, Statistics and Operations Research, Victoria University of Wellington, PO Box 600, Wellington 6140, New Zealand b

art ic l e i nf o

a b s t r a c t

Article history: Received 7 September 2014 Received in revised form 23 November 2014 Accepted 22 January 2015 Available online 17 February 2015

Particle swarm optimization (PSO) is a metaheuristic designed to find good solutions to optimization problems. However, when optimization problems are subject to noise, the quality of the resulting solutions significantly deteriorates, hence prompting the need to incorporate noise mitigation mechanisms into PSO. Based on the allocation of function evaluations, two opposite approaches are generally taken. On the one hand, resampling-based PSO algorithms incorporate resampling methods to better estimate the objective function values of the solutions at the cost of performing fewer iterations. On the other hand, single-evaluation PSO algorithms perform more iterations at the cost of dealing with very inaccurately estimated objective function values. In this paper, we propose a new approach in which hybrid PSO algorithms incorporate noise mitigation mechanisms from the other two approaches, and the quality of their results is better than that of the state of the art with a few exceptions. The performance of the algorithms is analyzed by means of a set of population statistics that measure different characteristics of the swarms throughout the search process. Amongst the hybrid PSO algorithms, we find a promising algorithm whose simplicity, flexibility and quality of results questions the importance of incorporating complex resampling methods into state-of-the-art PSO algorithms. & 2015 Elsevier B.V. All rights reserved.

Keywords: Particle swarm optimization Population statistics Noisy optimization problems Hybrid methods Resampling methods Single-evaluation methods

1. Introduction Particle swarm optimization (PSO) is a metaheuristic where a swarm of particles explores the search space of an optimization problem to find good solutions. Designed by Eberhart and Kennedy [1,2], it takes inspiration from swarming theory [3] and social models [4] by having particles interact with each other in order to improve the quality of their solutions. Each particle has a position that encodes a potential solution to the problem at hand, a velocity that will change the position of the particle at the next iteration, and a memory to remember where the particle found the best solution. Particles start at random positions and iteratively adjust their velocities such that they become partially attracted towards the positions of the best solutions found by themselves and their neighbors. At each step, particles evaluate their newly found positions and decide whether to store them in memory replacing previous findings. This is the regular PSO algorithm that has been

n

Corresponding author. E-mail addresses: [email protected] (J. Rada-Vilela), [email protected] (M. Johnston), [email protected] (M. Zhang). http://dx.doi.org/10.1016/j.swevo.2015.01.003 2210-6502/& 2015 Elsevier B.V. All rights reserved.

adapted to address many optimization problems in different fields of research [5–9]. In optimization problems subject to noise, the performance of PSO is an aspect that has not been as thoroughly studied as the performance of other metaheuristics like genetic algorithms [10–13] and evolution strategies [14,15]. In this type of problems, the objective function values that determine the quality of the solutions are corrupted by the effect of sampling noise, hence resulting in differently estimated objective function values every time the solutions are evaluated. As a consequence, particles eventually fail to distinguish good from bad solutions, leading to three conditions known as deception, blindness and disorientation [16]. Particles suffer from deception when they fail to select their true neighborhood best solutions, from blindness when they ignore truly better solutions, and from disorientation when they prefer truly worse solutions. The deterioration of the quality of the results found by PSO on optimization problems subject to noise prompts the need to incorporate noise mitigation mechanisms in order to prevent (or at least reduce) such a deterioration. Based on the use of the computational budget of function evaluations, the literature has distinguished two conceptually different approaches to mitigate the effect of noise on PSO. On the one hand, resampling-based PSO

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J. Rada-Vilela et al. / Swarm and Evolutionary Computation 22 (2015) 15–29

algorithms [17] allocate multiple function evaluations to the solutions in order to better estimate their objective function values by a sample mean over the evaluations [18–21]. On the other hand, single-evaluation PSO algorithms [22] do not allocate additional function evaluations to the solutions and focus instead on reducing the effect of having solutions with very inaccurately estimated objective function values [23–27]. Furthermore, since the computational budget is fixed and limited, resampling-based and single-evaluation PSO algorithms present opposite tradeoffs: resampling-based PSO algorithms better estimate the objective function values of the solutions at the cost of performing fewer iterations, whereas single-evaluation PSO algorithms perform more iterations at the cost of dealing with solutions whose objective function values are very inaccurately estimated. Recently, resampling-based and single-evaluation PSO algorithms have been studied by means of a set of population statistics that measure different characteristics of the swarms throughout the search process [16,17,22,28]. The population statistics have revealed that swarms often suffer from deception, blindness and disorientation, for which different algorithms have been designed to reduce the presence of these conditions and hence improve the quality of their resulting solutions. While previous works have focused exclusively on either resampling-based [17] or singleevaluation [22] PSO algorithms, in this article we perform a direct comparison between their population statistics. More importantly, in spite of the opposite tradeoffs of resampling-based and singleevaluation PSO algorithms, we merge their noise mitigation mechanisms into a new group of hybrid PSO algorithms. In doing so, we expect that the joint efforts of noise mitigation mechanisms in the new hybrid PSO algorithms will lead to a better quality of results than the purely resampling-based and single-evaluation PSO algorithms, respectively. The overall goal of this paper is to study the population statistics for new hybrid PSO algorithms and compare them against state-of-the-art resampling-based and single-evaluation PSO algorithms on optimization problems whose objective functions are subject to different levels of multiplicative Gaussian noise. Specifically, we will focus on the following objectives:

Merge noise mitigation mechanisms from single-evaluation

and resampling-based PSO algorithms into different hybrid PSO algorithms. Study the population statistics for the new hybrid PSO algorithms. Contrast the population statistics for the new hybrid PSO algorithms against the population statistics for the respective resampling-based and single-evaluation PSO algorithms. Contrast the population statistics of resampling-based and single-evaluation PSO algorithms.

The remainder of this paper is structured as follows. Section 2 provides some background on PSO, optimization problems subject to noise, population statistics for PSO, and related work. Section 3 presents the new group of hybrid PSO algorithms. Section 4 describes the design of experiments. Section 5 presents the results and discussions. Finally, Section 6 presents the conclusions and suggestions for future work.

2. Background 2.1. Particle swarm optimization Particle swarm optimization (PSO) is a metaheuristic designed by Kennedy and Eberhart [1,2] with inspiration from swarming theory [3] and social models [4]. It consists of a population of individuals that collectively explore the search space of an optimization problem to find good solutions. The population is referred to as a swarm, the individuals are referred to as particles, and the collective behavior results from the interactions between the particles. Specifically, each particle i consists of a position vector xti at iteration t that encodes a solution to the problem, a velocity vector vti to change xti in order to explore new solutions, and a memory vector yti to store the personal best position found. In addition, particle i belongs to a neighborhood of particles N i t from which the neighborhood best position y^ i is selected. The PSO algorithm is presented in Fig. 1 for a minimization problem, where f ðxÞ is the objective function value of the solution represented by position x. Usually, for each particle i, the position xi is initialized with random values sampled from a uniform distribution Uðxmin ; xmax Þ, where xmin and xmax are the boundaries of the optimization problem; the velocity vi is initialized to a null vector; and the position yi is initialized to an empty vector whose f ðyi Þ ¼ 1. Eqs. (1) and (2), vijt þ 1 and xtijþ 1 , refer to the values of velocity and position (respectively) of particle i at dimension j for the next iteration, w refers to the inertia coefficient [29], c1 and c2 are positive acceleration coefficients that determine the influence of the personal and neighborhood best positions, r t1j and r t2j are random values sampled from a uniform distribution Uð0; 1Þ, ytij is the value of dimension j of the personal best position found by t particle i, and y^ ij is the value of dimension j of the neighborhood best position selected by particle i. Hereinafter, we refer to the positions of the particles mostly as solutions. The network topology of the swarm defines the neighborhoods to which particles belong, thereby establishing links between the particles from which they can select their neighborhood best solutions. The most commonly used topologies are the ring and the star [30], but others have also been proposed in the literature [31]. On the one hand, the ring topology defines each neighborhood N i as the set of n particles adjacent to i, usually with n¼ 2. On the other hand, the star topology defines each neighborhood N i as the entire set of particles in the swarm, for which the star topology is equivalent to the ring topology when n ¼ j Sj . Therefore, the network topology influences the quality of the neighborhood best solutions as well as the diversity of the solutions in the swarm. Specifically, the ring topology encourages exploration as more particles are partially attracted towards different neighborhood solutions, whereas the star topology encourages exploitation as more particles are partially attracted towards the same neighborhood solutions [31–33]. 2.2. Optimization problems subject to noise

Fig. 1. Particle swarm optimization for a minimization problem.

Optimization problems subject to noise are a type of problem in which the objective function values of the solutions are corrupted by the effect of noise. As such, the objective function values of the


solutions are underestimated or overestimated, rarely reflecting their true values, and causing a significant deterioration to the performance of metaheuristics, in particular that of PSO [16]. In controlled environments, the effect of noise is usually modeled by sampling noise from a Gaussian distribution [34], whose standard deviation and effect on the objective function values determine the severity of noise. Typical effects of noise are additive and multiplicative, as shown in Eqs. (3) and (4), respectively, where f ðxÞ is the objective function value of solution x and Nðμ; σ 2 Þ is a random value sampled from a Gaussian distribution with mean μ and standard deviation σ. Hereinafter, we utilize the following terminology and notation: f ðxÞ is the true objective function value of solution x, f_ ðxÞ is a sampled objective function value of solution x, and f~ ðxÞ is the particle's estimated objective function value of solution x. f_ þ ðxÞ ¼ f ðxÞ þ N 0; σ 2 ð3Þ f_ ðxÞ ¼ f ðxÞ N 1; σ 2

ð4Þ

The effect of multiplicative noise produces more corruption on the objective function values than additive noise at the same standard deviation. More importantly, the effect of multiplicative noise changes proportionally to the objective function values, thereby producing a varying effect of noise throughout the search space. Such a varying effect has been characterized as the direction of the optimization problem [15], which is defined as backwards when the global optima is located towards the regions of low levels of noise, and forwards when the global optima is located towards the regions of high levels of noise. For example, a minimization problem with a positive objective function space is backwards when it is subject to multiplicative noise; whereas a maximization problem under the same conditions is forwards [16]. 2.3. Population statistics: Deception, blindness and disorientation The population statistics for PSO [16] measure different characteristics of a swarm with regards to its performance throughout the search process. In optimization problems subject to noise, the population statistics can measure the presence of three conditions from which particles suffer, namely deception, blindness and disorientation [16]. A particle suffers from deception when it fails to select the true neighborhood best solution, from blindness when it ignores a current solution that is truly better than its personal best solution, and from disorientation when it replaces its personal best solution with a truly worse current solution. Deception, blindness and disorientation are responsible for the deterioration of the quality of the results obtained with PSO on optimization problems subject to noise, and their respective population statistics are presented in Eqs. (5)–(7) for a swarm S after t max iterations as follows: 8 t max < 1 if y^ ti a ytω j ω ¼ arg minf ðytj Þ t t 1XX jANi I D^ ðSÞ ¼ D^ i ; where D^ i ¼ : jS ji A S t ¼ 1 0 otherwise ð5Þ (

I B ðSÞ ¼

t max 1XX Bt ; jS j i A S t ¼ 1 i

where Bti ¼

I D ðSÞ ¼

t max 1XX Dt ; jS ji A S t ¼ 1 i

where Dti ¼

1

if f ðxti Þ o f ðyti Þ

0

otherwise

(

1

if f ðyti Þ 4 f ðyti 1 Þ

0

otherwise

ð6Þ

ð7Þ

In addition, the population statistics also measure the cause of blindness and disorientation, which is attributable to the memory if the objective function value of the personal best solution is less accurately estimated than that of the current solution, and to the

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environment otherwise. Other population statistics that have shown to be helpful for the analysis of the effect of noise and the design of better noise mitigation mechanisms are found in [16,17,22] as follows. The effect of blindness and disorientation measures the average magnitude of hypothetical improvements missed by blindness and the average magnitude of deterioration caused by disorientation. The ranked deception measures the average true ranking of the selected neighborhood best solutions with respect to the swarm over the iterations performed, thereby complementing the information provided by the binary deception from Equation (5). The regular operation measures the average number of iterations at which particles correctly update their personal best solutions or correctly discard their current solutions, both of which are computed individually as regular updates and regular discards. The lifetime of the swarm measures the average iteration at which particles no longer find better solutions. La stly, the quality of results represents the set of true objective function values of the estimated best solutions found over all independent runs. 2.4. Related work Rada-Vilela et al. [16] identified and defined deception, blindness and disorientation as three conditions from which particles of PSO suffer when addressing optimization problems subject to noise. They designed a set of population statistics to measure the presence of these conditions in a swarm, alongside other characteristics throughout the search process. They computed the population statistics for the regular PSO and PSO with equal resampling (PSO-ER), each additionally under the assumptions of local and global certainty [18], on 20 large-scale benchmark functions whose objective function values are subject to different levels of multiplicative Gaussian noise. The population statistics revealed large proportions of deception and blindness affecting the particles of PSO and PSO-ER, confirmed the sensitivity of PSO to optimization problems subject to noise, highlighted the importance of utilizing resampling methods, and presented evidence of the higher priority of addressing blindness and disorientation before addressing deception. Bartz-Beielstein et al. [18] studied the performance of PSO on optimization problems subject to noise under the assumptions of local and global certainty. On the one hand, each particle in PSO with local certainty (PSO-LC) operates utilizing the true objective function values of its current and personal best solutions, and hence particles always have a regular operation as they discard worse solutions and update better solutions. On the other hand, each particle in PSO with global certainty (PSO-GC) operates utilizing the true objective function values of the personal best solutions from the particles within their neighborhoods, and hence particles always select their true neighborhood best solutions. PSO-LC was described as offering a superior quality of results due to its continuous progress throughout the optimization process and its resilience to noise and stagnation up to a certain extent. Conversely, PSO-GC was described as suffering from stagnation without converging to an optimum due to particles being misled by the effect of noise. While their study does not include quantitative results to support their analyses on local and global certainty, their findings on local and global certainty are supported in [16]. Rada-Vilela et al. [17] studied and compared the population statistics for the following resampling-based PSO algorithms: PSOER, which divides and equally allocates the computational budget amongst the current solutions; PSO with extended equal resampling (PSO-EER), which divides and allocates the computational budget amongst the current and personal best solutions, PSO with equal resampling and allocations to the top-N solutions (PSO-

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ERN), which divides and allocates the computational budget amongst a few good solutions and selects from them the neighborhood best solutions [35]; and PSO with optimal computing budget allocation (PSO-OCBA) [19], which sequentially allocates the computational budget amongst the current and personal best solutions to asymptotically minimize deception. The population statistics confirmed that PSO-ER finds the worst quality of results amongst the algorithms due to a large presence of deception and blindness. Two settings of PSO-EER find better results than PSO-ER thanks to an important reduction of deception and blindness in spite of an increment of deception. PSO-ERN finds better results than PSO-EER thanks to the selection of better neighborhood best solutions. PSO-OCBA finds the best quality of results amongst the algorithms thanks to the significantly smaller presence of deception achieved by a sequential reduction of blindness and disorientation. Pan et al. [19] proposed the incorporation of the optimal computing budget allocation (OCBA) resampling method [36,37] into PSO to asymptotically maximize the probability of correct selection of the neighborhood best solutions (i.e. minimize the probability of binary deception). PSO-OCBA sequentially allocates a computational budget at every iteration amongst the estimated best solutions in the swarm, hence improving the accuracy of the most relevant solutions that could actually be the true neighborhood best solution. In addition, they developed another algorithm based on PSO-OCBA to encourage the diversity of the swarm by means of hypothesis testing between the solutions, starting from the best solution to the worst solution. If the statistical difference between any two solutions is not significant, the worse solution is discarded and replaced with a new randomly generated solution. They compared PSO-OCBA, PSO-OCBA with hypothesis testing, and PSO-ER on six benchmark functions with two to six dimensions subject to additive Gaussian noise. The results showed that PSOOCBA with hypothesis testing finds better solutions than PSOOCBA, and the two find better solutions than PSO-ER. Further studies on PSO-OCBA and other resampling-based PSO algorithms can be found in [18,20,21,35,38,39]. Rada-Vilela et al. [22] studied and compared the population statistics for the following PSO algorithms: PSO with evaporation (PSO-E) [24], which encourages particles to forget their personal best solutions whenever they fail to find better solutions; PSO with probabilistic updates (PSO-PU), which encourages particles to probabilistically replace their personal best solutions whenever they fail to find better solutions; and PSO with average neighborhoods (PSO-AN), which computes a centroid solution from the swarm to be used as the neighborhood best solution. These algorithms estimate the objective function values with a single function evaluation, for which they refer to them as singleevaluation PSO algorithms. The population statistics revealed that particles in PSO-E suffer from too much disorientation, leading to divergent behavior and to the worst quality of results amongst the algorithms. PSO-PU provides better control and more exploitation than PSO-E, which reduces the presence of disorientation and improves the quality of the results. However, the regular PSO lacking a noise mitigation mechanism finds better solutions than PSO-E and PSO-PU thanks to a minimal presence of disorientation. The best quality of results in the presence of noise was obtained with PSO-AN thanks to its reduced disorientation and its reduced ranked deception achieved with the centroid solution. Cui et al. [24] proposed the evaporation mechanism for PSO to improve its performance on optimization problems subject to noise. PSO-E consists of worsening the estimated objective function values of the personal best solutions by an evaporation factor ρ whenever particles are unable to find better solutions. They utilized a PSO variant where particles store in memory a copy of the neighborhood best solutions, for which they also utilize

another evaporation factor ρ^ for these solutions. The experiments were performed on the three-dimensional sphere function whose solution space is subject to additive Gaussian noise with σ A f0:01; 0:025; 0:05g, hence classifying the problem as a search for robust solutions according to the classification in Jin and Branke [34]. In the absence of noise, the quality of the results obtained with PSO-E using ρ ¼ ρ^ ¼ 0:36 was not significantly different from that obtained with the regular PSO. However, in the presence of noise, the quality of the results obtained with PSOE was significantly better. Additionally, based on further experiments, they suggest that the best evaporation factors are ρ A ½0:44; 0:54 and ρ^ A ½0:27; 0:36. Further studies on evaporation and other single-evaluation PSO algorithms can be found in [23,25–27,40,41].

3. Hybrid methods in particle swarm optimization The literature has explored a variety of noise mitigation mechanisms to address deception, blindness and disorientation in PSO, but these mechanisms generally fall into two categories: resampling methods [17] and single-evaluation methods [22]. On the one hand, the population statistics for resampling-based PSO algorithms have shown that the best quality of results is obtained with PSO-OCBA [17], which is an algorithm designed to asymptotically minimize the presence of binary deception in the swarm. However, in spite of its design goal, particles in PSO-OCBA still suffer from a large presence of binary deception that extends (on average) to 78.53% of the iterations performed [17]. On the other hand, the population statistics for single-evaluation PSO algorithms have shown that the best quality of results is obtained with PSO-AN [22], which is an algorithm designed to reduce ranked deception by selecting centroid solutions instead of the estimated neighborhood best solutions. However, particles in PSOAN have shown to have a short lifetime that extends to 54.53% iterations, hence wasting almost half of the computational budget after that. Therefore, we expect that a hybrid PSO algorithm that incorporates a resampling method and a centroid solution will not only extend the lifetime of the swarm, but will also improve the quality of the neighborhood best solutions, both of which are positive indicators to a potentially better quality of results. The hybrid PSO algorithms that we propose consist of merging the best resampling methods for PSO with the centroid solution of PSO-AN. As such, the hybrid PSO algorithms differ from their purely resampling-based counterparts only on the selection of the neighborhood best solutions, and from PSO-AN on the better estimated objective function values of the solutions and the fewer iterations performed due to resampling. The remainder of this section describes the centroid solution and the algorithms that we propose. 3.1. The centroid solution The centroid solution is a single-evaluation method by which a particle computes an average solution from the personal best solutions within its neighborhood, and such a solution is selected by the particle to serve as its neighborhood best solution [22]. The goal of the centroid solution is to blur the effect of noise that hinders the selection of good solutions in the swarm (similar to the genetic repair effect in evolution strategies [42]). Specifically, when the objective function values of the solutions are very inaccurately estimated (like in single-evaluation PSO algorithms), the population statistics of ranked deception have shown that the true objective function value of the centroid solution is better (on average) than any other solution selected from the neigh borhood [22].


The centroid solution was proposed for the PSO-AN algorithm [22], which is described in Fig. 2 for a minimization problem. Originally, the centroid solution was computed utilizing Ci ¼ N i , but since the objective function values of the hybrid algorithms are better estimated thanks to a resampling method, the centroid solution that we propose is instead computed from a set Ci of the estimated best solutions within neighborhood N i . 3.2. The resampling method Resampling methods in PSO are noise mitigation mechanisms that perform multiple evaluations to a solution in order to better estimate its objective function value with a sample mean over the evaluations. Thus, the more evaluations performed to a solution, the better estimated its objective function value will be as its pffiffiffi standard error will reduce proportional to 1= n after n evaluations. However, since the additional evaluations performed are extracted from a fixed and limited computational budget B available to the algorithm, incorporating a resampling method into PSO creates a tradeoff between the number of re-evaluations and the number of iterations. The resampling methods that we incorporate into PSO divide the computational budget Bt at iteration t into two budgets, namely Btα and Btβ . The first computational budget Btα is equally divided and allocated between the current solutions in the swarm, and the second computational budget Btβ is allocated according to the resampling method. Thus, for example, PSO-ER equally allocates both Btα and Btβ between the current solutions, whereas PSOEER [17] equally allocates Btα between the current solutions and Btβ between the personal best solutions. Throughout this paper, we assume that the computational budget Bt is equal for every iteration t.

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the computational budget for the iteration is exhausted, the PSO algorithm continues its regular operation utilizing the means over the evaluations as the estimated objective function values of the solutions. Originally, the neighborhood best solution in PSO-OCBA was selected as the estimated best solution amongst the personal best solutions, but in the hybrid PSO-OCBA the centroid solution is utilized instead. The hybrid PSO-OCBA is described in Fig. 3, where bα is the number of function evaluations allocated to each current solution, X ti and Y ti are the sets of function evaluations allocated to the current and personal best solutions (respectively), Btβ is the computational budget of function evaluations to be allocated by OCBA, and bΔ is the number of function evaluations to be allocated at each step of OCBA. The operation of OCBA consists of dividing Btβ into groups of bΔ function evaluations, which are allocated according to the z-scores of the solutions with respect to the estimated neighborhood best solution [17]. The allocation of Btβ is determined according to Eqs. (8) and (9) in order to maximize the probability of correct selection of the neighborhood best solution [36]: ðe q e a Þ ðe p e a Þ pffiffiffiffiffi ¼ pffiffiffiffiffi ; sq = bq sp = bp

p; q A 1; 2; …; k and p a q a a

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2 k u X bp ba ¼ sa t 2 s p ¼ 1;p a a p

ð8Þ

ð9Þ

where bp is the number of function evaluations allocated to solution p, e p and sp are the mean and standard deviation over the set of function evaluations performed to solution p, a refers to the estimated best solution (i.e. the solution with the smallest mean), and k ¼ 2jS j is the number of (current and personal best) solutions in the swarm. The reader is encouraged to refer to [36,37] for further details on OCBA.

3.3. Hybrid PSO with optimal computing budget allocation 3.4. Hybrid PSO with extended equal resampling PSO-OCBA was proposed in Pan et al. [19] to asymptotically maximize the estimated probability for particles to correctly select the neighborhood best solution in a swarm fully connected by the star topology. In other words, PSO-OCBA was designed specifically to minimize the presence of binary deception within the swarm. PSO-OCBA incorporates the OCBA resampling method proposed in [36], which sequentially allocates evaluations to the solutions which are likely to be the true best solution within the swarm. The function evaluations are sequentially allocated amongst the current and personal best solutions according to their z-scores, hence favoring the allocation of function evaluations to the solutions whose estimated objective function values have smaller means and larger variances. After each group of function evaluations is allocated, the means and variances are updated and utilized to allocate the next group of function evaluations. Once

Fig. 2. PSO with average neighborhoods (adapted from [22]).

PSO with extended equal resampling (PSO-EER) was proposed in [17] as an extension to PSO-ER to reduce the large proportions of deception (92.74%) and blindness (36.13%) from which its particles suffer in optimization problems subject to noise [16]. In PSO-EER, the computational budget Btβ is equally divided and exclusively allocated amongst the personal best solutions, different from the PSO-ER which allocates it amongst the current solutions. As such, the objective function values of the personal best solutions are better estimated at each iteration, thereby successfully reducing the proportions of deception and blindness

Fig. 3. Hybrid PSO-OCBA.

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[17]. However, since the computational budget is fixed and limited, there is an accuracy tradeoff between the current and personal best solutions with respect to their estimated objective function values, that is, allocating more function evaluations to the personal best solutions requires to allocate proportionally fewer function evaluations to the current solutions, and vice versa. The population statistics computed for PSO-EER with one and two evaluations to the personal best solutions have shown that, while two evaluations further reduces deception and blindness, it produces too much disorientation and hence worse results than using a single evaluation to each personal best solution [17]. The hybrid PSO-EER algorithm is described in Fig. 4, where bα and bβ are the numbers of function evaluations allocated to the current and personal best solutions of each particle, respectively. 3.5. Hybrid PSO with equal resampling and allocations to the top-N solutions PSO with equal resampling and allocations to the top-N solutions (PSO-ERN) is an algorithm proposed in Rada-Vilela et al. [17] to be simpler and computationally cheaper than PSO-OCBA. The PSO-ERN algorithm allocates the computational budget Btβ at once between the estimated best N current solutions, unlike the sequential approach of PSO-OCBA. Particles in PSO-ERN decide to update their personal best solutions based on the means over the evaluations and on the number of evaluations performed to favor accuracy. As such, this two-criteria update rule encourages more frequent updates at the cost of increasing disorientation [17]. In addition, PSO-ERN attempts to reduce deception by selecting the neighborhood best solutions exclusively from those solutions with the most evaluations. Different from PSO-ERN, the hybrid PSO-ERN computes and selects the centroid solution as the neighborhood best solution. The hybrid PSO-ERN algorithm is presented in Fig. 5, where bi is the individual computational budget of function evaluations allocated from Btβ to particle i, and N is the number of current solutions whose objective function values will be more accurately estimated.

centroid solution from the estimated best five (personal best) solutions in the swarm. The choice of computing the centroid solution from only five solutions is arbitrary, and its purpose is to provide a starting point for future research. Hereinafter, we refer to these hybrid algorithms as PSO-EER(5), PSO-ERN(5) and PSOOCBA(5), where(5) indicates the number of solutions from which the centroid solution is computed. In addition, we want to compare the population statistics for hybrid PSO algorithms against those for single-evaluation and resampling-based PSO algorithms to assess their respective differences. Beside these comparisons, the population statistics between resampling-based and single-evaluation PSO algorithms will also be contrasted to determine which approach finds better results on optimization problems subject to noise. Thus, we will compare the population statistics for the following algorithms: PSO-EER [17], PSO-ERN Rada-Vilela et al. [17], PSO-OCBA Pan et al. [19], PSO-EER(5), PSOERN(5), PSO-OCBA(5), PSO-AN [22], and the PSO with equal resampling and global certainty (PSO-ERGC) [17] to serve as a baseline reference. The PSO-ER algorithm is excluded for comparison because PSO-EER has shown to be better [17]. The swarms of each algorithm are made up of 50 particles whose inertia w and acceleration coefficients c1 and c2 are set according to the guidelines in [43], and their neighborhoods are defined by the star topology. Particles limit their velocities utilizing the hyperbolic tangent function to reduce the sensitivity of setting a maximum velocity [31], which is set according to the limits xmin and xmax of the optimization problem. The computational budget available to each swarm is set to 30 000 function

4. Design of experiments The algorithms for which we want to compute the population statistics are the hybrid PSO algorithms made up from different resampling methods and using the centroid solution as the neighborhood best solution. Specifically, we will focus on PSOEER, PSO-ERN and PSO-OCBA, each of which computing the Fig. 5. Hybrid PSO-ERN.

Table 1 Parameter settings. Parameter

Value

Independent runs Number of particles

50 with 30 000 function evaluations

Acceleration Inertia Maximum velocity Velocity clamping

Fig. 4. Hybrid PSO-EER.

50 in R1000 with star topology Static with c1 ¼ c2 ¼ 1:49618 Static with w¼0.729844 0:25 xmax xmin ! vtij v_ tij ¼ vmax tanh vmax

Severity of noise

σ A f0:06; 0:12; 0:18; 0:24; 0:30g

PSO-EER, PSO-EER(5) PSO-OCBA, PSO-OCBA(5) PSO-ERN, PSO-ERN(5)

bα ¼ 5; bβ ¼ 1 bα ¼ 5; bΔ ¼ 5 bα ¼ 5; N ¼ 2; bð1Þ ¼ bð2Þ ¼ 25


21

algorithms. Lastly, Section 5.7 presents four new hybrid PSO algorithms and the discussions of their respective population statistics. The four new hybrid algorithms are based on PSO-AN and PSO-EER to allocate differently the computational budget of function evaluations.

evaluations, thereby allowing the swarms of PSO-AN to perform 600 iterations, whereas resampling-based and hybrid PSO algorithms perform 100 iterations. As such, the population statistics for PSO-AN are made up from 50 600 50 ¼ 1 500 000 observations and those for the remaining algorithms are made up from 50 100 50 ¼ 250 000 observations. The parameter values of PSO-OCBA are selected based on the suggestions in [37,36], and those of PSO-EER and PSO-ERN are selected according to our findings in [17]. In general, the settings are the same as those in [16,17,22], but with a different set of algorithms. The complete list of parameter settings is presented in Table 1. The population statistics will be computed for the different algorithms on the optimization problems presented at the CEC'2010 Special Session and Competition on Large-Scale Global Optimization [44]. This suite of benchmarks presents 20 largescale minimization functions whose objective function values are only positive and the global minimum is f ðxÞ ¼ 0. According to the degree of separability, these benchmark functions are classified into five sets as shown in Table 2, where the parameters d ¼ 1000 and m ¼50 refer to the number of dimensions and the size of the groups (respectively). These benchmarks are composed (primarily) of the classical optimization functions shown in Table 3. In particular, the most challenging functions are rastrigin and ackley because the former has numerous local minima surrounding the global minimum and the latter has little gradient information to find the global minimum [16]. The benchmark functions are converted into backward optimization problems by corrupting the objective function values with multiplicative Gaussian noise. The algorithms are set to perform 50 independent runs on each benchmark function at diffe rent levels of noise ranging from very low to very high as σ A f0:06; 0:12; 0:18; 0:24; 0:30g. At each level of noise, the noise samples are ensured to lie within 3σ (by resampling, if needed) in order to keep the objective function values only positive and remove the complexity of dealing with extreme outliers [16]. Specifically, since 99.73% of the samples of the Gaussian distribution lie within 3σ [45], our truncation method will only apply to the remaining 0.27% of the samples, thereby discarding samples in the range ð 1; 3σ Þ and ð3σ ; 1Þ. As such, the true objective function value of each solution will be corrupted at most by a factor of 1:0 73σ . For example, at σ ¼0.30, the objective function value of a solution will be corrupted by up to 90% of its true objective function value.

Our discussions are based mostly on the average population statistics presented in Table 4 utilizing the knowledge from our previous findings in [16,17,22]. Additionally, we present in Fig. 6 the population statistics computed on F13 to provide a visual representation of a concrete benchmark that follows the average trend. The population statistics for each benchmark function are presented in the Appendices. The algorithms in Fig. 6 are abbreviated as (e) PSO-EER, (n) PSO-ERN, (o) PSO-OCBA, (E) PSO-EER(5), (N) PSO-ERN(5), (O) PSO-OCBA(5), (a) PSO-AN and (g) PSO-ERGC, and the population statistics are presented as follows [16,17,22]. Quality of results. The boxplots represent the true objective function values (left axis) of the best solutions found by the algorithms (bottom axis) on F13 subject to the different levels of noise (top axis). The boxplots are colored from light to dark gray to ease the comparison. The benchmark functions are minimization problems and, therefore, lower objective function values indicate better solutions. Binary deception. The barplots represent the average proportion of iterations (left axis) at which a particle of each algorithm (bottom axis) is deceived by its neighbors on F13 subject to the different levels of noise (bars colored from light to dark gray). Smaller proportions are better. Particles from PSO-ERGC do not suffer from binary deception as they always select their true neighborhood best solutions. Ranked deception. The barplots represent the normalized average rank (left axis) of the neighborhood best solutions with respect to the swarm of each algorithm (bottom axis) on F13 subject to the different levels of noise (bars colored from light to dark gray). Smaller ranks are better as they indicate better neighborhood best solutions. Particles from PSO-ERGC do not suffer from ranked deception as they always select their true neighborhood best solutions. Regular operation, blindness and disorientation. The stacked barplots represent the average proportions (left axis) of regular operation (dark gray), blindness (medium gray) and disorientation (light gray) experienced by a particle of each algorithm (bottom

5. Results and discussions

Table 3 Composition of the benchmark functions.

The results and discussions are structured as follows. Section 5.1 presents and describes the population statistics for the algorithms. Section 5.2 compares resampling-based and hybrid PSO algorithms. Section 5.3 compares the hybrid PSO algorithms. Sections 5.4 and 5.5 compare the hybrid PSO algorithms against the resampling-based PSO-OCBA and PSO-ERGC, respectively. Section 5.6 compares the single-evaluation PSO-AN against resampling-based and hybrid PSO

5.1. Population statistics

Composition

A

B

C

D

E

elliptic rastrigin ackley schwefel rosenbrock

F01 F02 F03 – –

F04 F05 F06 F07 F08

F09 F10 F11 F12 F13

F14 F15 F16 F17 F18

– – – F19 F20

Source: Tang et al. [44].

Table 2 Separability of the benchmark functions. Set

Functions

Description

A B C D E

½F 01–03 ½F 04–08 ½F 09–13 ½F 14–18 ½F 19–20

Separable, each dimension can be independently optimized from the others. Partially separable, only a single group of m dimensions are non-separable. Partially separable, d=2m groups of m dimensions are non-separable. Partially separable, d=m groups of m dimensions are non-separable. Fully non-separable, any two dimensions cannot be optimized independently.

Source: Tang et al. [44].

22


Table 4 Average population statistics for the PSO algorithms over all benchmark functions, levels of noise, independent runs, iterations and particles. Statistic

Lifetime þ B. Deception R. Deception R. Operation þ Updates þ Discards Blindness Memory Environment Disorientation Memory Environment

Resampling

Hybrid

EER

ERN

OCBA

EER(5)

ERN(5)

OCBA(5)

93.58 88.39 30.78 69.84 24.40 75.60 22.31 68.33 31.67 7.85 15.50 84.50

88.89 86.22 23.45 60.87 25.10 74.90 33.53 81.19 18.81 5.60 24.61 75.39

90.47 78.53 20.89 67.60 23.12 76.88 27.33 71.75 28.25 5.07 12.00 88.00

95.19 87.20 21.21 57.74 35.71 64.29 34.65 72.13 27.87 7.62 17.61 82.39

87.72 89.44 27.54 56.85 27.00 73.00 37.75 83.48 16.52 5.41 25.49 74.51

91.43 83.30 15.99 56.18 31.34 68.66 39.21 74.35 25.65 4.61 10.49 89.51

Larger þ or smaller is better. Units in percentages (%). Bold value is best in group. Boldvalue is best overall.

F13

σ06

σ18

σ12

σ30

σ24

1.5e+12

5.0e+11 e n o E NO a g e n o E NO a g e n o E NO a g e n o E NO a g e n o E NO a g

1.0

F13

1.0

0.5

0.0

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0.5

e

n

o

E

N

O

a

g

F13

1.0

1.0

e

n

o

E

N

O

a

g

F13

0.0

1.0

n

o

E

N

O

a

g

n

o

E

N

O

a

g

n

o

E

N

O

a

g

n

o

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e

F13

0.5

e

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O

a

g

F13

0.0

1.0

0.5

0.0

e

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0.5

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F13

e

F13

0.5

e

n

o

E

N

O

a

g

0.0

e

Fig. 6. Population statistics on benchmark function F13 (details in Section 5.1, p. 20).

AN

ERGC

54.43 97.64 31.77 41.99 3.59 96.41 57.64 99.00 1.00 0.36 5.00 95.00

72.47 (0.00) (0.00) 53.04 27.31 72.69 44.75 89.17 10.83 2.21 12.42 87.58


Table 5 Summary of statistical tests on the quality of the results between resampling-based and hybrid PSO algorithms. EER(5) vs. EER

¼

þ

σ06 σ12 σ18 σ24 σ30 Total ERN(5) vs. ERN σ06 σ12 σ18 σ24 σ30 Total OCBA(5) vs. OCBA σ06 σ12 σ18 σ24 σ30 Total

17 16 16 17 17 83

3 4 4 3 3 17

0 0 0 0 0 0

15 15 13 15 13 71

4 5 7 5 7 28

1 0 0 0 0 1

17 14 15 15 14 75

2 4 4 5 6 21

1 2 1 0 0 4

axis) on F13 subject to the different levels of noise (bars from left to right). Larger proportions of regular operations and smaller proportions of blindness and disorientation are better. Regular updates and discards. The stacked barplots represent the average proportions (left axis) of regular updates (dark gray) and discards (light gray) experienced by a particle of each algorithm (bottom axis) on F13 subject to the different levels of noise (bars from left to right). Larger proportions of regular updates and smaller proportions of regular discards are better. Causes of blindness. The stacked barplots represent the average proportions of blindness (left axis) caused by memory (dark gray) and by the environment (light gray) in a particle of each algorithm (bottom axis) on F13 subject to the different levels of noise (bars from left to right). Causes of disorientation. The stacked barplots represent the average proportions of disorientation (left axis) caused by memory (dark gray) and by the environment (light gray) in a particle of each algorithm (bottom axis) on F13 subject to the different levels of noise (bars from left to right). Effect of disorientation and blindness. The stacked barplots represent the normalized average magnitudes (left axis) of deterioration caused by disorientation (dark gray) and hypothetical improvements missed by blindness (light gray) on F13 subject to the different levels of noise (bars from left to right). Lifetime. The barplots represent the normalized average lifetime (left axis) of a particle of each algorithm (bottom axis) on F13 subject to the different levels of noise (bars colored from light to dark gray). A longer lifetime is better. The population statistics in Fig. 6 and Table 4 are discussed in the following sections and are referred to by their names in italics the first time that are brought up for discussion. In addition, summaries of statistical tests on the quality of the results between two algorithms on all the benchmark functions are presented at each level of noise, where the values indicate the number of benchmark functions on which the difference of the quality of the results from the first algorithm is significantly better ( ) or worse (þ ) than the second algorithm, or not statistically significant (¼ ), according to the pairwise Wilcoxon test at α ¼ 0.05 with Holm correction. For the sake of brevity, we have removed the “PSO-” abbreviation of the algorithms from the tables hereinafter (e.g. PSO-EER is presented as EER).

23

5.2. Resampling-based algorithms and hybrid algorithms The quality of results obtained with the hybrid PSO algorithms is significantly better than that of their purely resampling-based counterparts, and that is generally the case as shown in Table 5. The underlying reasons to such an improved quality of results are explained utilizing the remaining population statistics as follows. The ranked deception significantly decreases by about 10% in PSO-EER(5), thus showing that the centroid solution is more often better than the estimated neighborhood best solution. Even when PSO-EER(5) decreased the ranked deception by a larger proportion than PSO-OCBA(5) did (both with respect to their resamplingbased counterparts), PSO-OCBA(5) still has a smaller ranked deception (15.99% vs. 21.21%) because its centroid solution is made up from better solutions. In PSO-ERN(5), differently, the binary and ranked deception each increases by more than 3% and 4% with respect to PSO-ERN. Such a difference is expected because PSO-ERN allocates its additional computational budget between the estimated best two solutions, leaving many other solutions without additional evaluations. Hence, the centroid solution will be made up utilizing less accurately estimated solutions which may not even include the two solutions whose objective function values are the most accurate. A workaround to this issue would be to compute the centroid solution from the solutions with the most evaluations, which will not necessarily be the solutions with the estimated best objective function values. Nonetheless, the quality of the results of PSO-ERN(5) is still much better than that of PSO-ERN, even with such a deterioration in binary and ranked deception. In PSO-OCBA(5), the binary deception increases by about 5% with respect to PSO-OCBA, thus showing that the centroid solution is more often worse than the true neighborhood best solution. However, the ranked deception reduces by about 5% in PSOOCBA(5), which indicates that the centroid solution is more often better than the estimated neighborhood best solution. The better ranked deception of PSO-OCBA(5) favorably compensates for its worse binary deception in the long term, specifically because particles from both PSO-OCBA and PSO-OCBA(5) fail very often to select the true neighborhood best solutions. The proportions of blindness significantly increase by 12% in PSO-EER(5) and PSO-OCBA(5), and by 4% in PSO-ERN(5), all with respect to their purely resampling-based counterparts, even when the only difference between them is the selection of the centroid solution. However, this is not unexpected since previous works have reported that selecting better neighborhood best solutions increase the presence of blindness in the swarm [16,17]. Specifically, the most compelling cases can be found in the comparisons between PSO-ERGC and PSO-ER in [16,17], where blindness increased by over 8% on average in PSO-ERGC with respect to PSO-ER. The other population statistics show that the centroid solution helps to increase the proportions of regular updates in PSO-EER(5) and PSO-OCBA(5) by more than 8% and 11%, and just by 2% in PSOERN(5), all with respect to their purely resampling-based counterparts. 5.3. Hybrid algorithms The quality of results shows that PSO-OCBA(5) finds better solutions than PSO-EER(5), and PSO-EER(5) finds better solutions than PSO-ERN(5). This ranking is further supported by the transitive relation found in the summary of statistical tests in Table 6. Partly, the ranking is influenced by the centroid solution computed for the algorithms and its effect on the binary and ranked deception (as discussed in Section 5.2). Moreover, since the proportions of regular operations in the hybrid algorithms are

24


rather similar, the proportions of regular updates and regular discards alone serve as indicators of the superior performance of PSO-EER(5) and PSO-OCBA(5) over PSO-ERN(5). In addition, other statistics that partly support the ranking between the hybrid algorithms are the larger proportions of blindness by memory and disorientation by memory in PSO-ERN(5), both of which are detrimental characteristics that indicate that the objective function values of the personal best solutions in PSO-ERN(5) are less accurately estimated than those in PSO-EER(5) and PSO-OCBA(5). 5.4. Hybrid algorithms and PSO-OCBA PSO-EER(5) and PSO-ERN(5) are hybrid algorithms whose operation is simpler and computationally less expensive than that of PSO-OCBA, and yet both hybrids still find significantly better solutions in the majority of the benchmark functions as shown in Table 7. PSO-EER(5) and PSO-ERN(5) are the first algorithms in the literature to outperform the resampling-based PSO-OCBA in so many benchmark functions, therefore questioning the importance of the large efforts put towards correctly selecting the neighborhood best solutions in the swarms. The comparison hereinafter is made between the population statistics for PSO-EER(5) and PSO-OCBA. We only focus on this comparison because we already compared PSO-OCBA(5) and PSOOCBA in Section 5.2, and because PSO-ERN(5) yields the worst quality of results amongst the hybrid algorithms. Different from our previous works that explain the quality of the results based on the population statistics [16,17,22], the goal of this comparison is to find out whether we could predict the better quality of the results obtained with PSO-EER(5) by just looking at the population statistics. The lifetime of PSO-EER(5) is about 5% longer than that of PSO-OCBA (95.19% vs. 90.47%), which is generally a positive indicator for a superior quality of results. However, the lifetime needs to be considered together with the proportions of regular operations, and especially the proportions of regular updates. In regular operations, the proportions of PSO-OCBA are larger by about 10% than those of PSO-EER(5) (67.60% vs. 57.74%). In regular updates, however, the proportions of PSO-OCBA are smaller by about 12% than those of PSO-EER(5) (23.12% vs. 35.71%). Given the conflicting statistics of regular operations and regular updates between PSO-EER(5) and PSO-OCBA, and considering that the regular updates are computed as proportions of the regular operations, we can solve the conflict by computing the proportions of global regular updates with respect to the number of iterations as: 0:3571 57:74 ¼ 20:62% for PSO-EER(5), and 0:2312 67:60 ¼ 15:63% for PSO-OCBA. Therefore, on average,

Table 6 Summary of statistical tests on the quality of the results between hybrid PSO algorithms. OCBA(5) vs. EER(5)

¼

þ

σ06 σ12 σ18 σ24 σ30 Total

5 8 6 7 10 36

15 11 14 13 10 63

0 1 0 0 0 1

EER(5) vs. ERN(5) σ06 σ12 σ18 σ24 σ30 Total

11 13 13 12 13 62

9 7 7 8 7 38

0 0 0 0 0 0

PSO-EER(5) not only has a longer lifetime, but it also has about 5% more iterations with regular updates, which supports the superior quality of the results of PSO-EER(5) over PSO-OCBA. The binary deception for PSO-OCBA is smaller by about 9% than PSO-EER(5) (78.53% vs. 87.20%). However, the proportions of binary deception in both algorithms are still very large, which indicates that the algorithms seldom select the true neighborhood best solutions. Thus, the ranked deception provides more information about the swarm given that it measures the average ranking of whatever neighborhood best solution that is selected. In ranked deception, the difference between PSO-OCBA and PSO-EER(5) is just about 1% favoring PSO-OCBA. Therefore, given such a negligible difference, the superior quality of results expected from PSOEER(5) still holds thanks to its longer lifetime and larger proportions of global regular updates. The proportions of blindness and disorientation, however, are about 7% and 3% smaller for PSO-OCBA than they are for PSO-EER(5) (27.33% vs. 34.65%, and 5.07% vs. 7.62%), thereby contradictorily suggesting that the quality of the results obtained with PSO-OCBA should be better. Certainly, we cannot attribute the larger proportions of blindness to the selection of better neighborhood best solutions (as we have done before) because both algorithms have rather similar proportions of ranked deception, and we cannot dismiss the small differences in the proportions of disorientation because we have seen before the detrimental effects of disorientation on the quality of the results [22]. Therefore, we can only advise to take the population statistics as very useful guidelines of performance having in mind that contradictions are still possible. These findings do not discredit the importance of the population statistics designed thus far to analyze the performance of PSO algorithms, but encourage instead the creation of additional population statistics to represent other variables throughout the search process that could be helpful to better estimate the quality of the results. 5.5. Hybrid algorithms and PSO-ERGC The previous comparison between hybrid algorithms and the resampling-based PSO-OCBA questioned the relevance of making costly efforts towards the correct selection of the neighborhood best solutions, especially considering that the efforts made by PSO-OCBA to such an end still fail on average in 78.53% of the iterations (see Table 4). More importantly, if PSO-OCBA successfully managed to correctly select the true best solution at every iteration, that is, if PSO-OCBA were as successful as PSO-ERGC, the hybrid algorithms would still find better solutions in most cases as shown in Table 8. The population statistics for the hybrid algorithms suggest their superior quality of results against PSO-ERGC. Specifically, the hybrid algorithms have a longer lifetime, more regular operations divided into more regular updates (and hence fewer regular discards), and smaller proportions of blindness. These statistics compensate for the absence of binary and ranked deception in PSOERGC as well as for its smaller proportions of disorientation. 5.6. Single-evaluation against resampling-based and hybrid algorithms In Table 4, the average population statistics for PSO-AN are significantly worse than those for the resampling-based and hybrid PSO algorithms. The only exception is the population statistic of disorientation, where PSO-AN shows the smallest (and hence best) average value due to the presence of very large proportions of blindness. Contradictorily, the summary of statistical tests in Table 9 shows that the quality of the results obtained with PSO-AN is not necessarily worse, but actually significantly better in about half of the benchmark functions. Specifically, PSO-AN finds better solutions than


Table 7 Summary of statistical tests on the quality of the results between hybrid PSO algorithms and PSO-OCBA. EER(5) vs. OCBA

¼

þ


16 14 13 13 12 68

1 4 4 5 6 20

3 2 3 2 2 12

ERN(5) vs. OCBA σ06 σ12 σ18 σ24 σ30 Total

15 10 10 11 10 56

1 7 6 5 2 21

4 3 4 4 8 23

the resampling-based PSO algorithms in about half of the benchmark functions regardless of the level of noise, and worse solutions in the remaining half with a few cases in which the differences are not significant. Similarly, PSO-AN finds better solutions than the hybrid PSO algorithms in about half of the benchmark functions, but only at medium-to-high levels of noise. However, observing the data from which the summary of statistical tests is computed, Figs. 1–4 in the Online Supplementary Material show that PSO-AN is only better on the benchmark functions that are composed of ackley and rastrigin, and on two benchmark functions composed of rosenbrock. These remarkable differences suggest that the appropriate accuracy tradeoff is problem dependent. Therefore, in the next section, we will explore the effect of utilizing different allocations of the computational budget on the population statistics and the quality of the results. 5.7. Budget allocation: from PSO-AN to PSO-EER(5) Besides the remarkable differences found in the quality of results between PSO-AN and the hybrid algorithms, the population statistics for PSO-AN in Table 4 show that the average lifetime of PSO-AN is 54.43%, which suggests that about half of the iterations performed are a waste of function evaluations which could have been better spent on re-evaluating every (current or personal best) solution at least once. In doing so, not only the objective function values would have been better estimated, but also the proportions of blindness could have been significantly reduced and the regular operations increased, alongside other potential positive effects such as a longer lifetime and more regular updates in spite of a likely increase of disorientation. Furthermore, once the (current or personal best) solutions are better estimated, the centroid solution can be created utilizing only a few of the estimated best solutions instead of all of the solutions, thereby reducing the binary and (mostly) the ranked deception in the swarm. With these expectations in mind, we performed an additional set of experiments utilizing PSO-AN and hybrid variants of PSO-AN that spend the computational budget differently. Specifically, we designed the following four hybrid PSO-AN algorithms: PSO-AN11, PSO-AN21, PSO-AN31, and PSO-AN41, where the subscripts are in the form xy and refer to the number of evaluations performed to the current solution (x) and to the personal best solution (y). As such, these hybrid variants cover the range of different computational budget allocations between the single-evaluation PSO-AN and the resampling-based PSO-EER. The new hybrid algorithms balance differently the accuracy tradeoff, for which they perform the following number of iterations: 600 in PSO-AN, 300 in PSO-AN11, 200 in PSO-AN21, 150 in PSO-AN31, 120 in PSO-AN41, and

25

100 in PSO-EER (or PSO-AN51, equivalently). Furthermore, the new algorithms are hybridized by utilizing centroid solutions computed from the estimated best five solutions in the swarm. The design of experiments and the parameters utilized are the same as those described in Section 4 and Table 1. Besides these hybrids, we include PSO-OCBA(5) to serve as a reference. The quality of the results obtained with PSO-AN and its hybrid variants on F13 are shown in Fig. 7, where the algorithms are abbreviated as follows: (a) PSO-AN, (1–5) PSO-ANx1, and (O) PSOOCBA(5). The average population statistics for the set of algorithms over all the benchmark functions are shown in Table 10. Fig. 7 confirms our expectations of improving the quality of the results of PSO-AN by re-evaluating the current and personal best solutions. Furthermore, the quality of the results highlights the importance of sacrificing iterations over improving the accuracy of the estimated objective function values. PSO-EER (or PSO-AN51, equivalently) shows the best accuracy tradeoff, and the trends observed in Fig. 7 are also similar to those on the remaining benchmark functions, which were excluded for the sake of brevity. Moreover, the average population statistics in Table 10 confirm our expectations on the lifetime of the swarms (binary and ranked) deception, regular operations, blindness and disorientation, all of which are discussed in the following paragraphs. However, we found again exceptions in ackley and rastrigin, and we discuss them at the end of this section. The lifetime of PSO-AN11 is the longest amongst the algorithms, while the lifetime of PSO-AN is the shortest, and in between the lifetime progressively shortens as the computational budget allocates more function evaluations to the current solutions. First, the longest lifetime of PSO-AN11 highlights the importance of allocating function evaluations to the personal best solutions in order to reduce blindness and thereby prevent stagnation to what may not even be a local optimum. PSO-AN11 has the longest lifetime thanks to a continuous correction of the estimated objective function values of the personal best solutions and to the very inaccurately estimated objective function values of the current solutions. Specifically, the continuous correction will reduce the proportions of blindness, but will also ease the replacement of the personal best solutions by the current solutions. Therefore, the personal best solutions are more easily replaced throughout the search, which explains the longer lifetime and also the important proportions of disorientation. Differently, the lifetime of the other hybrid PSO-AN algorithms shortens because the current solutions are better estimated, which makes harder to find a better solution once they replace the personal best solutions. The proportions of binary and ranked deception reduce as more function evaluations are allocated to the current solutions. Since current solutions eventually become personal best solutions, better estimating their objective function values would favor a selection of better neighborhood best solutions in the swarm. However, notice that the binary deception in PSO-AN is only slightly larger than that of PSO-AN11, whereas its ranked deception is smaller by about 7%. We attribute the smaller ranked deception of PSO-AN to its shorter lifetime and its larger number of iterations as follows. A lifetime of 54.43% suggests that the personal best solutions do not change much in the last half of the search process, for which we also expect that the ranking of the neighborhood best solution will not change much either. If we consider the candidates for neighborhood best solution being sorted according to their true objective function values, we expect that the ranking of the estimated neighborhood best solution will lead to a ranking biased towards the better half of the solutions. Hence, considering both lifetime and number of iterations, PSO-AN can actually have a better ranked deception. Overall, algorithms PSO-AN11 to PSOAN51 show that the more function evaluations allocated to the current solutions, the better estimated their objective function

26


Table 8 Summary of statistical tests on the quality of the results between hybrid PSO algorithms and PSO-ERGC. EER(5) vs. ERGC

¼

þ


16 11 11 11 10 59

0 5 5 5 6 21

4 4 4 4 4 20

ERN(5) vs. ERGC σ06 σ12 σ18 σ24 σ30 Total

12 10 8 8 8 46

4 5 7 4 3 23

4 5 5 8 9 31

ERGC vs. OCBA σ06 σ12 σ18 σ24 σ30 Total

13 15 13 13 14 68

7 5 7 7 6 32

0 0 0 0 0 0

values are, and hence the binary and ranked deception are reduced once the current solutions become personal best solutions. Compared to PSO-AN, the proportions of blindness in PSO-AN11 significantly reduce by about 25% thanks to the allocation of a single function evaluation to the personal best solutions. However, blindness remains mostly unchanged as more function evaluations are allocated to the current solutions (i.e. from PSO-AN11 to PSOAN51), which must be due to the fewer iterations that the algorithms perform. Regardless, blindness is still reduced, which in turn increases the proportions of regular operations as would be expected from solutions with better estimated objective function values. Furthermore, as more function evaluations are allocated to the current solutions, the proportions of disorientation also reduce thanks to the current solutions having better estimated objective function values. While the inverse correlation between blindness and disorientation [16] is not clear in these cases, we attribute it to the different number of iterations that the algorithms perform. The proportions of blindness by the environment decrease thanks to the current solutions having better estimated objective function values and to the backward direction of the optimization problem [16], leading consequently to a proportional increase of blindness by memory. Similarly, the proportions of disorientation by the environment increase mostly because the direction of the optimization problem is backwards, and hence the probability density functions of sampling noise in worse solutions will generally have larger standard deviations than those in their previous (and better) personal best solutions [16]. As the disorientation by the environment increases, the disorientation by memory proportionally decreases. Lastly, in most of the benchmark functions based on ackley and rastrigin, the quality of the results showed that PSO-AN finds better solutions than any of the new hybrid algorithms, and yet the average population statistics still follow the same general trends that we have discussed before. The population statistics on these exceptions fail to provide further insights on what makes PSO-AN produce a better quality of results than the hybrid algorithms, for which we decided to omit them. However, these findings do not discredit the importance of the population statistics to analyze the performance of PSO algorithms on optimization problems subject to noise, but rather encourage further studies to create other statistics that could be more helpful to better predict the quality of the results. Overall, these findings

still provide useful information for future studies to address the relative importance of the different population statistics with respect to the quality of the results. 5.8. Further discussions The population statistics for PSO on optimization problems subject to noise comprises a set of studies [16,17,22,28] that we have pursued in order to systematically reduce the deterioration of the quality of the results caused by the effect of noise. The population statistics for PSO were first presented in [16], where we found that particles of the regular PSO and PSO-ER suffer from deception, blindness and disorientation in the presence of noise, for which addressing these conditions we expected to improve the quality of the resulting solutions. Subsequently, we proceeded to study the population statistics for resampling-based [17] and single-evaluation [22] PSO algorithms. On the one hand, the population statistics for resamplingbased PSO algorithms [17] presented two new algorithms that performed better than PSO-ER, namely PSO-EER and PSO-ERN, but the best performing algorithm was the PSO-OCBA proposed in [19]. On the other hand, the population statistics for single-evaluation PSO algorithms presented the PSO-PU as a new algorithm that performed better than the evaporation mechanism in PSO-E, but still the regular PSO and especially the PSO-AN performed better than PSO-PU and PSO-E. The population statistics from these studies provided important insights about the presence of deception, blindness and disorientation, and their role in the deterioration of the quality of the results. Furthermore, such studies motivated the exploration of the hybrid methods presented here, whose performance is significantly better than that of resampling-based and single-evaluation PSO algorithms in most cases.

6. Conclusions and future work Particle swarm optimization is a metaheuristic whose performance is sensitive to optimization problems subject to noise as particles suffer from three conditions, namely deception, blindness and disorientation, which are responsible for the deterioration of the quality of the resulting solutions. The incorporation of noise mitigation mechanisms into PSO aims to reduce the presence of these three conditions throughout the search process in order to prevent (or at least reduce) such a deterioration. Based on the allocation of function evaluations performed by the noise mitigation mechanisms, we distinguish three groups of PSO algorithms as: resampling-based [17], which sacrifice performing more iterations over better estimating the objective function values; single-evaluation [22], which sacrifice the accuracy of the objective function values over performing more iterations; and a new group of hybrids, which merge methods from resampling-based and single-evaluation. Hybrid PSO algorithms combine resampling and singleevaluation methods to reduce deception, blindness and disorientation in the swarm. While it is possible to create a variety of different hybrid PSO algorithms, we focused on combining the resampling methods studied in [17] and the best performing single-evaluation method designed in [22]. As such, we created three new hybrid algorithms based on PSO-EER, PSO-ERN and PSO-OCBA, each combined with the centroid solution utilized in PSO-AN. Since the new algorithms better estimate the objective function values of the solutions thanks to their respective methods, the centroid solutions are computed instead from five of the estimated best solutions in the swarm and not from all the solutions as PSO-AN does. The goal of restricting the centroid to be computed from a few solutions is to prevent worse solutions from blurring the effect of better solutions, whereas such is not the

3 4 8 8 8 31 17 11 11 11 10 60 3 3 8 8 9 31

12 11 10 10 10 53

þ

4 0 1 0 0 5

¼

4 9 9 10 10 42

2 0 0 3 2 7


8 10 10 10 11 49 σ06 σ12 σ18 σ24 σ30 Total AN vs.

¼

0 6 1 1 1 9

¼ EER(5) ERN(5)

10 10 10 10 7 47 2 0 0 0 2 4 8 10 10 10 11 49 10 10 10 7 7 44

þ ¼ þ

0 4 1 1 1 7

¼ þ

12 10 10 10 10 52 0 1 0 0 0 1 8 9 10 10 10 47

¼

þ

OCBA(5)

8 8 8 9 9 42

ERGC OCBA EER ERN AN vs.

Table 9 Summary of statistical tests on the quality of the results of PSO-AN against resampling-based and hybrid PSO algorithms.

17 12 11 11 11 62

þ

12 12 11 11 11 57 0 0 1 0 0 1

¼

þ


27

case of PSO-AN where the objective function values of all the solutions are very inaccurately estimated per se. The hybrid algorithms found significantly better solutions than their purely resampling-based counterparts in most of the cases, and in the remaining cases the solutions were of similar quality with only a few exceptions in which the quality was actually worse. The main reason for such improvements was the reduced (binary and ranked) deception in the swarms thanks to the use of centroid solutions. However, as a consequence of selecting the centroid solutions as better neighborhood best solutions, we observed again an important increase of the proportions of blindness. Such a particular correlation between deception and blindness has been observed before in [16,17], especially during the comparisons between PSO-ER against PSO-ERGC, but the underlying reasons remain uncertain. Amongst the hybrid algorithms, PSO-OCBA(5) finds the overall best quality of results, although PSO-EER(5) is still a strong competitor as its difference in quality of results with respect to PSO-OCBA(5) is not statistically significant in most cases. The superior quality of results obtained with PSO-OCBA(5) is mostly thanks to the better estimated objective function values of the most important solutions, which leads to a better selection of solutions from which the centroid solutions are computed. However, PSO-EER(5) is still a promising algorithm due to the quality of its results and, more importantly, due to its simpler approach that may prove more convenient to derive new algorithms from. Regarding PSO-ERN(5), not only the quality of its results is worse than that of its hybrid counterparts, but the tractability of its results is also harder to analyze due to its two-criteria replacement of the personal best solutions. While we find no compelling reason to further investigate the performance of PSO-ERN(5), we find that the twocriteria replacement of the personal best solutions is a cheap strategy to increase the diversity of the solutions in the swarm. The vastly superior quality of results found by hybrid PSO algorithms does not fully extend to the case of the singleevaluation PSO-AN. Even when the hybrid PSO algorithms find significantly better solutions in most cases, PSO-AN still manages to outperform the hybrid algorithms in about half of the benchmark functions at high levels of noise, most of which are made up of ackley and rastrigin functions. These results suggest that our research has reached a point where it is difficult for a single algorithm to outperform others on every problem addressed, and hence the most appropriate balance of the accuracy tradeoff needs to be adjusted specifically to the problem at hand in order to exploit some of its inherent characteristics. Considering the comparison between single-evaluation and hybrid PSO algorithms, it is not unexpected to observe PSO-AN outperforming the purely resampling-based PSO algorithms in the same problems. In fact, the quality of the results obtained with PSO-AN is significantly better than that of the resampling-based PSO algorithms in about half of the benchmark functions regardless of the level of noise, and significantly worse in the other half. Again, such a superior quality of results is found mostly on the benchmarks made up of ackley and rastrigin, but also of rosenbrock in two cases. Such a clear segmentation on the quality of the results obtained with single-evaluation and resampling-based PSO algorithms further supports that, hereinafter, the most appropriate balance of the accuracy tradeoff is dependent on the characteristics of the problem at hand. The accuracy tradeoff is determined by the allocation of the computational budget of function evaluations. Thus, in order to explore different accuracy tradeoffs on the problems addressed, we designed four new hybrid PSO-AN algorithms that allocate the computational budget differently. Each of the four new algorithms gradually moved from PSO-AN to PSO-EER, that is, from allocating a single function evaluation to every current solution to allocating multiple function evaluations to every current solution and an

28


F13

σ06

σ18

σ12

σ30

σ24

8e+11

4e+11 a 1 2 3 4 5Oa 1 2 3 4 5Oa 1 2 3 4 5Oa 1 2 3 4 5Oa 1 2 3 4 5O Fig. 7. Quality of results. The boxplots represent the true objective function values (left axis) of the best solutions found by acPSO-AN, its hybrid variants and PSO-OCBA(5) (bottom axis) at each level of noise (top axis) in all independent runs on benchmark function F13. The boxplots are colored from light to dark gray to ease the comparison. The benchmark functions are minimization problems, therefore lower objective function values indicate better solutions.

Table 10 Average population statistics for PSO-AN and hybrid algorithms over all benchmark functions, levels of noise, independent runs, iterations and particles. Statistic

AN

AN11

AN21

AN31

AN41

AN51

OCBA(5)

Lifetime þ B. Deception R. Deception

54.43 97.64 31.77

99.05 97.08 38.56

98.25 94.55 30.91

97.32 92.15 26.38

96.30 89.52 23.28

95.19 87.20 21.21

91.43 83.30 15.99

R. Operation þ Updates þ Discards

41.99 3.59 96.41

52.34 38.33 61.67

53.94 35.33 64.67

55.33 34.63 65.37

56.61 35.02 64.98

57.74 35.71 64.29

56.18 31.34 68.66

Blindness Memory Environment

57.64 99.00 1.00

31.71 51.49 48.51

33.84 61.75 38.25

34.63 66.96 33.04

34.78 70.09 29.91

34.65 72.13 27.87

39.21 74.35 25.65

Disorientation Memory Environment

0.36 5.00 95.00

15.96 26.62 73.38

12.22 22.79 77.21

10.04 20.10 79.90

8.61 18.53 81.47

7.62 17.61 82.39

4.61 10.49 89.51

Larger þ or smaller is better. Boldvalue is the best. Units in percentages (%).

additional computational budget between the personal best solutions. The quality of the results obtained with these algorithms gradually improved as they approached PSO-EER. However, on the cases of ackley and rastrigin, PSO-AN is still better. Even when the population statistics of the new algorithms matched our expectations, the quality of the results was not improved as we had expected. Nonetheless, we consider these findings important to encourage further studies that could attempt to better estimate the quality of the results based on the population statistics. Further research on this topic could address the following objectives:

Design hybrid algorithms that incorporate resampling methods

and other dissipative mechanisms such as evaporation or probability updates [22]. Study the effect of computing the centroid solutions from different numbers of estimated best solutions, as well as the effect of weighted centroid solutions utilizing their objective function values as weights. Design hybrid heterogeneous PSO algorithms whose particles utilize different noise mitigation mechanisms that could be statically selected for the search or dynamically during the search [46–50]. Design additional population statistics that could provide further information about the swarm throughout the search process. Study the population statistics for hybrid PSO algorithms on forward optimization problems and problems subject to different noise models. Explore the characteristics of ackley and rastrigin that hinder the performance of the PSO algorithms in order to

design better noise mitigation mechanisms that specifically exploit such characteristics.

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