otrA inFormAción

This chapter has presented an overview of EC with an emphasis on SI, which is the class of algorithms to which PSO belongs given its source of inspiration. Moreover, a comprehensive overview of optimization prob- lems subject to noise has been covered and distinguished from other op- timization problems according to the type of uncertainty present, thereby clearly delimiting the scope of the problems addressed in this thesis. The relevance of noisy optimization problems has been explored in the field of simulation-based optimization, where solutions to real-world problems are optimized based on evaluations that contain stochastic components due to simulation models. In this type of problem, the performance of PSO has been acknowledged to present two issues, namely the inaccurate memories of the particles and their most likely incorrect selection of the neighborhood best solutions. These challenges make the effect of noise on PSO inherently different from that on other metaheuristics such as GAs and ES. First, particles store in memory solutions with inaccurately esti- mated objective values, and that will affect the frequency at which they find better solutions. Second, particles select the neighborhood best so- lutions from the solutions stored in memory, for which poor decisions made by an individual particle will potentially affect the decisions of those within the neighborhood.

The noise mitigation mechanisms incorporated into PSO can be clas- sified based on whether they perform additional evaluations to the so- lutions to better estimate them. If they perform additional evaluations, we refer to them as resampling-based PSO algorithms, and several works have been explored in the literature suggesting PSO-OCBA is the best al- gorithm available to deal with this type of problem. Conversely, if the

2.9. SUMMARY 61 PSO algorithm does not perform any additional evaluation, we classify such an approach as a single-evaluation method given that the estimated objective values of the solutions are based solely on the single evaluation that the PSO algorithm dictates. Several works have also explored the performance of single-evaluation methods, all of which make use of the evaporation mechanism to worsen the quality of the personal best solu- tions and thereby encourage their replacement. While both resampling and single-evaluation methods have shown improvements on the quality of the results, their experimental design has not been too rigorous and, more often than not, their conclusions end up based mostly on intuition supporting hypothetical expectations rather than confirming them based on empirical results.

Lastly, considering the few benchmark functions upon which previous works have evaluated the performance of their algorithms, this chapter has presented the 20 large-scale benchmark functions on which we will evaluate the algorithms in this thesis. The benchmark functions make up the entire suite presented at the CEC’2010 Special Session and Competi- tion on Large-Scale Global Optimization [141], but have been presented with a different notation aiming to provide a less verbose mathematical representation than the one originally proposed.

Chapter 3

Deception, Blindness and

Disorientation

This chapter identifies and clearly defines deception, blindness and disorien- tation as three conditions responsible for the deterioration of the quality of the results of PSO on optimization problems subject to noise. Addi- tionally, this chapter introduces the concept of population statistics to mea- sure the extents to which these conditions (and other characteristics) affect the particles in the swarm throughout the search process. The population statistics are first studied for the regular PSO and PSO-ER, both addition- ally under the assumptions of local and global certainty.

This chapter is structured as follows. Section 3.1 introduces this chap- ter. Section 3.2 presents the population statistics and the definitions of deception, blindness and disorientation. Section 3.3 describes the design of experiments. Section 3.4 presents the results and discussions. Finally, Section 3.5 ends this chapter with a summary.

3.1

Introduction

One characteristic that has remained largely unexplored in PSO is the de- terioration of the quality of its results on optimization problems subject

to noise. In this type of problem, the objective values of the solutions are corrupted by the effect of sampling noise, thus causing solutions to have inaccurate objective values that change every time these are evaluated. As a consequence, particles eventually fail to distinguish good from bad solu- tions, leading in turn to other issues that ultimately end up deteriorating the quality of the results. The literature has recognized two such issues as particles having inaccurate memories and particles failing to select of their true neighborhood best solutions, both of which have been addressed re- spectively by incorporating evaporation mechanisms [30, 31, 32, 50, 51] and resampling methods [7, 106, 113, 154] into PSO.

The incorporation of noise mitigation mechanisms into PSO has shown to provide significant improvements on the quality of the results [7, 30, 31, 32, 50, 51, 106, 113, 154]. However, the analyses behind such improve- ments often fall short of empirical evidence supporting the claims made about the underlying reasons for their achievements in terms other than the quality of the results. More importantly, there is not even evidence showing the extents to which inaccurate memories affect the particles in the swarm or the frequency at which particles fail to select their true neigh- borhood best solutions. Therefore, given the lack of such evidence, we want to investigate the effect of noise on PSO beyond the deterioration of its results. Furthermore, we expect that such information will not only help to choose the best noise mitigation mechanism, but will also help to design even better ones.

In this chapter, we distinguish the effect of noise on particles as decep- tion when they fail to select their true neighborhood best solutions, blind- ness when they miss out on opportunities to improve upon their own best solutions, and disorientation when they mistakenly prefer worse solutions. While deception is just a new name to refer to the incorrect selection of neighborhood best solutions, blindness and disorientation are new con- cepts that clearly define the problem of inaccurate memories. Further- more, we design in this chapter a set of population statistics to count the

3.2. POPULATION STATISTICS FOR PSO 65