This chapter develops the first hybrid PSO algorithms based on the best single-evaluation method found in Chapter 4, namely PSO-AN, and the different resampling methods studied in Chapter 5. The population statis-tics for the new algorithms are studied on the set of benchmark functions subject to noise, and compared against the population statistics for the purely single-evaluation and resampling-based PSO algorithms. Further-more, the effect of different computational budget allocations is explored on the population statistics for PSO-AN and other hybrid variants based on PSO-AN and PSO-EER.
This chapter is structured as follows. Section 6.1 introduces this chap-ter. Section 6.2 presents the design of new hybrid PSO algorithms. Sec-tion 6.3 describes the design of experiments. SecSec-tion 6.4 presents the re-sults and discussions. Finally, Section 6.5 ends this chapter with a sum-mary.
6.1 Introduction
Noise mitigation mechanisms for PSO can be classified into two different approaches according to their allocation of the computational budget of function evaluations. On the one hand, there is single-evaluation methods
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that do not re-evaluate any solution, but instead settle with their objective values being estimated by a single evaluation as the PSO algorithm dic-tates. On the other hand, there is resampling methods which make use of the computational budget of function evaluations to perform multiple evaluations to the solutions in order to better estimate their true objective values. Hence, both approaches provide opposite tradeoffs regarding the exploration of new solutions and improving the accuracy of existing solu-tions.
The best single-evaluation method for PSO, amongst the ones we have studied, is PSO-AN. The goal of PSO-AN is to blur the effect of noise on the selection of the neighborhood best solution in order to reduce decep-tion. Each particle in PSO-AN computes the centroid from all the per-sonal best solutions in the neighborhood, and the centroid is selected as the neighborhood best solution. In doing so, the centroid solution will generally have a better quality than any other solution a particle would have selected based on the very inaccurately estimated objective values.
Consequently, such a better neighborhood best solution will have parti-cles partially attracted towards better regions of the search space that will improve the ranked deception and, ultimately, the quality of the results.
The best resampling method for PSO, amongst the ones we have stud-ied, is PSO-OCBA. The goal of PSO-OCBA is to asymptotically minimize the binary deception in the swarm. First, PSO-OCBA equally allocates the computational budget between the particles in order to better esti-mate their current solutions. Then, an additional computational budget is sequentially allocated to the most important (current or personal best) solutions, that is, those solutions whose estimated objective values have the lower means and larger variances. After each allocation of a group of evaluations, the means and variances of the objective values of the re-cipient solutions will be re-estimated and taken into consideration before allocating the next group of evaluations. Hence, after each allocation, the estimated best solutions will be reaffirmed or discarded to be selected as
6.1. INTRODUCTION 179 neighborhood best solutions. At the end, when the computational bud-get is exhausted, the solution with the estimated best objective value will potentially be the true neighborhood best solution.
While single-evaluation and resampling-based PSO algorithms are com-pletely different from each other, this does not prevent us from merging them and deriving a hybrid PSO algorithm. On the contrary, such a hy-brid algorithm could potentially deliver better results than both their al-gorithms independently from each other. The most compelling reason to support our argument is that PSO-OCBA still fails most of the time to cor-rectly select the true best solution and, even when it has a superior quality of results amongst resampling-based algorithms, selecting the centroid of the estimated best solutions will likely have a better objective value than that of the estimated best solution. Therefore, we expect that a hybrid be-tween PSO-AN and PSO-OCBA will reduce the proportions of (at least ranked) deception and thereby improve the quality of the solutions. More importantly, since this is not an exclusive case for PSO-OCBA, we can also hybridize PSO-EER with PSO-AN and possibly improve the quality of the results.
Chapter Goals
The overall goal of this chapter is to create hybrid algorithms between PSO-AN and different resampling-based methods as an attempt to fur-ther improve the quality of the results found with PSO on optimization problems subject to noise. Specifically, the following objectives will be ad-dressed:
• Contrast the population statistics for hybrid PSO algorithms with their respective single-evaluation and resampling-based PSO algo-rithms to determine whether hybrid methods find better solutions.
• Contrast the population statistics for hybrid PSO algorithms with
PSO-ERGC in order to determine the extent to which efforts should be made to correctly select the neighborhood best solutions.
• Contrast the population statistics for single-evaluation and resampling-based PSO algorithms in order to determine which approach finds better solutions.
6.2 Hybrid PSO Algorithms
Thus far, we have studied the population statistics for PSO utilizing single-evaluation and resampling methods, both of which balance the accuracy tradeoff differently. On the one hand, PSO with single-evaluation meth-ods perform more iterations at the cost of having the objective values of the solutions very inaccurately estimated. On the other hand, PSO with resampling methods sacrifice the number of iterations over improving the accuracy of the solutions. In each method, the algorithms that have shown the best quality of results are PSO-AN and PSO-OCBA. Hence, in order to further improve the performance of PSO, we incorporate into resampling-based PSO algorithms the idea of using, as neighborhood best solutions, the centroid computed from a number of estimated best solutions.
The centroid solution is utilized in the single-evaluation PSO-AN, where particles compute the centroid from the personal best solutions in the borhood, and use that to create a new solution that serves as the neigh-borhood best. As such, particles mitigate the effect of noise that causes deception by creating a potentially better solution that does not need to be evaluated for the algorithm to work. However, have in mind that we do evaluate the solution for the purpose of estimating its quality with respect to the swarm and thus compute the population statistics of deception.
The centroid solution was proposed in Chapter 4 as the average neigh-borhood best solution in PSO-AN, and the quality of results obtained was significantly better than that of any other single-evaluation algorithm