The closest alternative to PSO-OCBA is PSO-ERN, which is a simpler and computationally cheaper algorithm that manages to produce similar results in about half of the benchmark functions and worse results in the remaining benchmarks. PSO-ERN allocates the computational budget of additional evaluations between the estimated top-N solutions at once and not sequentially, thereby saving some computational time. Moreover, its particles update their personal best solutions with new solutions if these are better or more accurate, thus encouraging frequent updates. While PSO-ERN does not outperform PSO-OCBA, the quality of its results is bet-ter than PSO-ER and betbet-ter than both settings of PSO-EER, thus remarking the importance of having accurate objective values on at least a few good solutions.
The population statistics have shown again to be useful to understand the underlying reasons for the quality of the results of resampling-based PSO algorithms, and also helped to design the PSO-EER algorithm that finds better solutions than PSO-ER by reducing blindness and deception.
Next Chapter
The next chapter will present a study on the population statistics for hy-brid PSO algorithms that combine both single-evaluation and resampling methods to address optimization problems subject to noise. The popula-tion statistics for single-evaluapopula-tion and resampling-based PSO algorithms will also be compared to determine which approach (if any) produces bet-ter results.
5.A Population Statistics
The following figures correspond to the population statistics on each bench-mark function for the algorithms utilized in the design of experiments from this chapter (Section 5.3, page 150).
Sets A and E Set B
Figure 5.4: Quality of Results. The boxplots represent the true objective values (left axis) of the best solutions found by the algorithms (bottom axis) at each level of noise (top axis) in all independent runs. The al-gorithms are abbreviated as (g) PSO-ERGC, (e) PSO-ER, (1) PSO-EER1, (2) PSO-EER2, (n) PSO-ERN, and (o) PSO-OCBA. The boxplots are coloured from light to dark gray to ease the comparison. The benchmark functions are minimization problems, therefore lower objective values in-dicate better solutions. The boxplots in F06corresponding to (o) and (1) at σ06 have been upscaled as well as those corresponding to (g) in order to improve their presentation without changing their relative ordering.
5.A. POPULATION STATISTICS 169
Figure 5.5: Quality of Results. The boxplots represent the true objective values (left axis) of the best solutions found by the algorithms (bottom axis) at each level of noise (top axis) in all independent runs. The al-gorithms are abbreviated as (g) PSO-ERGC, (e) PSO-ER, (1) PSO-EER1, (2) PSO-EER2, (n) PSO-ERN, and (o) PSO-OCBA. The boxplots are coloured from light to dark gray to ease the comparison. The benchmark functions are minimization problems, therefore lower objective values in-dicate better solutions.
Sets A and E Set B Set C Set D
Figure 5.6: Binary Deception. The barplots represent the average propor-tions of iterapropor-tions (left axis) at which a particle is deceived by its neigh-bors for each algorithm (bottom axis) on the benchmark functions sub-ject to levels of noise σ ∈ {0.06, 0.12, 0.18, 0.24, 0.30} (bars coloured from light to dark gray). The algorithms are abbreviated as (g) PSO-ERGC, (o) PSO-OCBA, (n) PSO-ERN, (e1) PSO-EER1, (e2) PSO-EER2, and (e) PSO-ER. Particles in PSO-ERGC do not suffer from deception. Smaller proportions are better.
5.A. POPULATION STATISTICS 171
Figure 5.7: Ranked Deception. The barplots represent the average per-centile rank (left axis) based on the true objective values of the selected neighborhood best solutions with respect to the true objective values of the personal best solutions in the swarm for each algorithm (bot-tom axis) on the benchmark functions subject to levels of noise σ ∈ {0.06, 0.12, 0.18, 0.24, 0.30} (bars coloured from light to dark gray). The al-gorithms are abbreviated as (g) PSO-ERGC, (o) PSO-OCBA, (n) PSO-ERN, (e1) PSO-EER1, (e2) PSO-EER2, and (e) PSO-ER. Particles from PSO-ERGC do not suffer from deception and hence their neighborhood best solutions are always ranked best. Lower percentile ranks are better.
Sets A and E Set B Set C Set D
Figure 5.8: 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) ex-perienced by a particle for each algorithm (bottom axis) on the bench-mark functions subject to levels of noise σ ∈ {0.06, 0.12, 0.18, 0.24, 0.30}
(bars from left to right). The algorithms are abbreviated as (g) PSO-ERGC, (o) PSO-OCBA, (n) PSO-ERN, (e1) PSO-EER1, (e2) PSO-EER2, and (e) PSO-ER. Larger proportions of regular operation and smaller propor-tions of blindness and disorientation are better.
5.A. POPULATION STATISTICS 173
Figure 5.9: Regular Updates and Discards. The stacked barplots repre-sent the average proportions (left axis) of regular updates (dark gray) and discards (light gray) experienced by a particle for each algorithm (bottom axis) on the benchmark functions subject to levels of noise σ ∈ {0.06, 0.12, 0.18, 0.24, 0.30} (bars from left to right). The algorithms are ab-breviated as (g) PSO-ERGC, (o) PSO-OCBA, (n) PSO-ERN, (e1) PSO-EER1, (e2) PSO-EER2, and (e) PSO-ER. Larger proportions of regular updates and smaller proportions of regular discards are better.
Sets A and E Set B Set C Set D
Figure 5.10: Causes of Blindness. The stacked barplots represent the av-erage proportions (left axis) of blindness caused by memory (dark gray) and by the environment (light gray) in a particle for each algorithm (bot-tom axis) on the benchmark functions subject to levels of noise σ ∈ {0.06, 0.12, 0.18, 0.24, 0.30} (bars from left to right). The algorithms are ab-breviated as (g) PSO-ERGC, (o) PSO-OCBA, (n) PSO-ERN, (e1) PSO-EER1, (e2) PSO-EER2, and (e) PSO-ER.
5.A. POPULATION STATISTICS 175
Figure 5.11: Causes of Disorientation. The stacked barplots represent the average proportions (left axis) of disorientation caused by memory (dark gray) and by the environment (light gray) in a particle for each algorithm (bottom axis) on the benchmark functions subject to levels of noise σ ∈ {0.06, 0.12, 0.18, 0.24, 0.30} (bars from left to right). The algo-rithms are abbreviated as (g) PSO-ERGC, (o) PSO-OCBA, (n) PSO-ERN, (e1) PSO-EER1, (e2) PSO-EER2, and (e) PSO-ER.
Sets A and E Set B Set C Set D
Figure 5.12: Lifetime. The barplots represent the proportions of average lifetime (left axis) of a particle for each algorithm (bottom axis) on the benchmark functions subject to levels of noise σ ∈ {0.06, 0.12, 0.18, 0.24, 0.30} (bars coloured from light to dark gray). The al-gorithms are abbreviated as (g) PSO-ERGC, (o) PSO-OCBA, (n) PSO-ERN, (e1) PSO-EER1, (e2) PSO-EER2, and (e) PSO-ER. A longer lifetime is better when the swarm does not converge to the global optimum.