Population statistics for particle swarm optimization: Single-evaluation methods in noisy optimization problems

Abstract

Particle swarm optimization (PSO) is a metaheuristic whose quality of results deteriorates significantly in optimization problems subject to noise. The underlying reason to such a deterioration is that the effect of noise hinders the ability of particles to distinguish good from bad solutions, leading them to suffer from deception, blindness and disorientation. A deceived particle is not partially attracted to the true best solution in its neighborhood, a blinded particle misses an opportunity to improve upon its personal best solution, and a disoriented particle mistakenly prefers a worse solution. These conditions need to be addressed via noise mitigation mechanisms to prevent (or at least reduce) such a deterioration. Single-evaluation methods are the name by which we refer to PSO algorithms that address the effect of noise without performing additional function evaluations. The first of these algorithms was PSO with evaporation (PSO-E), which was proposed to reduce blindness in the swarms, and reports have suggested that it succeeds at finding better solutions than the regular PSO in different stochastic and dynamic optimization problems. However, PSO-E depends on an evaporation factor whose value is determined empirically, and the swarm is always at risk of exhibiting divergent behaviour. In this article, we propose a method to determine a priori the evaporation factor for PSO-E, and we also propose a new PSO with probabilistic updates (PSO-PU) to prevent the risk of divergence. Additionally, we take a different approach and develop a new PSO with average neighborhoods (PSO-AN) to blur the effect of noise and thereby reduce deception. Experiments on 20 large-scale benchmark functions subject to different levels of noise show that the regular PSO (lacking a noise mitigation mechanism) generally finds better solutions than PSO-E and PSO-PU because their approaches cause too much disorientation. However, PSO-AN finds better solutions than the regular PSO thanks to the improved quality of its neighborhood best solutions that partially attract the swarm towards better regions of the search space.

Appendices

Appendix A: Quality of results—sets $\mathcal {A}$, $\mathcal {E}$, $\mathcal {B}$

See Fig. 8.

Fig. 8

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 algorithms are abbreviated as (e) PSO-LE, (E) PSO-HE, (p) PSO-LP, (P) PSO-HP, (a) PSO, and (n) PSO-AN. The boxplots are coloured from light to dark gray to ease the comparison. The benchmark functions are minimization problems; therefore, lower objective values indicate better solutions

Appendix B: Quality of results—sets $\mathcal {C}$, $\mathcal {D}$

See Fig. 9.

Fig. 9

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 algorithms are abbreviated as (e) PSO-LE, (E) PSO-HE, (p) PSO-LP, (P) PSO-HP, (a) PSO, and (n) PSO-AN. The boxplots are coloured from light to dark gray to ease the comparison. The benchmark functions are minimization problems; therefore, lower objective values indicate better solutions

Appendix C: Regular operation, blindness and disorientation

See Fig. 10.

Fig. 10

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 for each algorithm (bottom axis) on the benchmark functions subject to levels of noise $\sigma \in \{0.06, 0.12, 0.18, 0.24, 0.30\}$ (bars from left to right). The algorithms are abbreviated as (e) PSO-LE, (E) PSO-HE, (p) PSO-LP, (P) PSO-HP, (a) PSO, and (n) PSO-AN. Larger proportions of regular operation and smaller ones of blindness and disorientation are better

Appendix D: Updates in regular operation

See Fig. 11.

Fig. 11

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 for each algorithm (bottom axis) on the benchmark functions subject to levels of noise $\sigma \in \{0.00, 0.06, 0.12, 0.18, 0.24, 0.30\}$ (bars from left to right). The algorithms are abbreviated as (e) PSO-LE, (E) PSO-HE, (p) PSO-LP, (P) PSO-HP, (a) PSO, and (n) PSO-AN. Larger proportions of regular updates are better

Appendix E: Causes of blindness

See Fig. 12.

Fig. 12

Causes of blindness. The stacked barplots represent the average proportions (left axis) of blindness caused by memory (dark gray) and the environment (light gray) in a particle for each algorithm (bottom axis) on the benchmark functions subject to levels of noise $\sigma \in \{0.06, 0.12, 0.18, 0.24, 0.30\}$ (bars from left to right). The algorithms are abbreviated as (e) PSO-LE, (E) PSO-HE, (p) PSO-LP, (P) PSO-HP, (a) PSO, and (n) PSO-AN

Appendix F: Causes of disorientation

See Fig. 13.

Fig. 13

Causes of disorientation. The stacked barplots represent the average proportions (left axis) of disorientation caused by memory (dark gray) and the environment (light gray) in a particle for each algorithm (bottom axis) on the benchmark functions subject to levels of noise $\sigma \in \{0.00, 0.06, 0.12, 0.18, 0.24, 0.30\}$ (bars from left to right). The algorithms are abbreviated as (e) PSO-LE, (E) PSO-HE, (p) PSO-LP, (P) PSO-HP, (a) PSO, and (n) PSO-AN. Particles from the regular PSO and PSO-AN do not suffer from disorientation in the absence of noise

Appendix G: Effect of disorientation and blindness

See Fig. 14.

Fig. 14

Effect of disorientation and blindness. The stacked barplots represent the average proportions of weighted magnitudes (left axis) of deterioration caused by disorientation (dark gray) and hypothetical improvements missed by blindness (light gray) on the benchmark functions subject to the levels of noise $\sigma \in \{0.00, 0.06, 0.12, 0.18, 0.24, 0.30\}$ (bars from left to right). The algorithms are abbreviated as (e) PSO-LE, (E) PSO-HE, (p) PSO-LP, (P) PSO-HP, (a) PSO, and (n) PSO-AN. Particles from the regular PSO and PSO-AN do not suffer from blindness or disorientation in the absence of noise

Appendix H: Optimization curves

See Fig. 15.

Fig. 15

Optimization curves. The plots represent the average objective values (left axis) of the personal best solutions over all the independent runs at each iteration (bottom axis) of PSO-LE (marked with bullet) and PSO-LP (no marks) on the benchmark functions subject to levels of noise $\sigma _{00}$ (solid line), $\sigma _{06}$ (long-dashed line), $\sigma _{18}$ (short-dashed line), and $\sigma _{30}$ (dotted line). The benchmark functions are minimization problems; therefore, lower objective values indicate better solutions