PSO with and without Resampling

The quality of results shows that PSO in the absence of noise finds bet- ter solutions than any of the other algorithms in the presence of noise. However, in the presence of noise, PSO-ER is significantly better than PSO even when it performs 100 iterations compared to the 600 that PSO per- forms. Nonetheless, both algorithms deteriorate in the presence of noise and higher levels of noise lead to further deterioration. These results show the sensitivity of PSO to optimization problems subject to noise and the importance of improving the accuracy of the objective values in order to provide a significantly better quality of results. The underlying reasons for such a difference are mostly found within the following statistics.

The lifetime of PSO is generally shorter than that of PSO-ER by about 10%, thus indicating that particles in PSO tend to stagnate earlier than those in PSO-ER. We attribute such a shorter lifetime partly to the larger proportions of blindness in PSO that prevent particles from updating to better solutions, and partly to the larger number of iterations performed by PSO which renders its convergence more likely. Notice that the dif-

σ00 σ06 σ12 σ18 σ24 σ30 a a l g e l g a l g e l g a l g e l g a l g e l g a l g e l g 5.0e+10 2.5e+11 F01 Quality of Results a e al el ag eg 0.0 0.5 1.0F01 Lifetime a e al el ag eg 0.0 0.5 1.0 F01 Deception a e al el ag eg 0.0 0.5 1.0F01

R. Operation, Blindness and Disorientation

a e al el ag eg

0.0

0.5

1.0F01

Regular Updates and Discards

a e al el ag eg 0.0 0.5 1.0F01 Causes of Blindness a e al el ag eg 0.0 0.5 1.0F01 Causes of Disorientation

Figure 3.1: Population statistics on benchmark function F01. The presenta-

3.4. RESULTS AND DISCUSSIONS 77 ference in lifetime between both algorithms is expressed in proportions, and as such, it means that particles in PSO spend a much larger number of iterations without finding better solutions. Moreover, as the level of noise becomes higher, the lifetime in both algorithms further shortens due to the increasing proportions of blindness. In general, the lifetime of the swarms suggests that they reach convergence far before performing 100% of the iterations, thereby supporting our choice of setting the computa- tional budget to 30 000 function evaluations.

The proportions of deception in both algorithms are very high (gener- ally over 95%), but those in PSO-ER are slightly smaller (over 85%). While we expected PSO-ER to present a much smaller proportion of deception because the objective values of its solutions are more accurate, these re- sults show that particles require even more re-evaluations to better esti- mate their objective values and hence increase the chance of correctly se- lecting the neighborhood best solutions. Nonetheless, the greater accuracy of PSO-ER does lead particles to generally select better neighborhood solu- tions than those in PSO, and this can be confirmed by utilizing a deception indicator measuring ranks instead of the binary selection presented here.

The proportions of regular operations are larger in PSO-ER (over 50%) than in PSO (below 50%) because the estimated objective values are more accurate in the former. Consequently, particles in PSO-ER are more likely to correctly distinguish good from bad solutions, and thereby correctly either update their personal best solutions or discard their current solu- tions. In the absence of noise, particles in PSO do not suffer from blindness or disorientation and hence all of them present only regular operations. However, as the level of noise increases, the proportions of regular opera- tions in PSO-ER progressively reduce as expected, but those in PSO show subtle differences with a small increase at σ = 0.30. While we would need to study in more detail the reasons for such a sudden increase of regular operations in PSO, the subtle differences just indicate that very low lev-

higher levels of noise will not have a major influence except for the quality of the results which deteriorates significantly.

The proportions of regular updates and discards in the absence of noise show that particles in PSO fail to find better solutions in over 50% of the iterations performed. Such a high proportion of failure (without finding the global optimum solution) is due to the exploration mechanism deter- mined by the velocity update in Equation (2.3) and its respective clamping. This suggests that it would be useful to explore the performance of other mechanisms such as the constriction factor proposed in [26] or design new mechanisms including more information about the problem upon updat- ing the velocity (e.g. proportionally weighting the personal and neighbor- hood best solutions according to their objective values). The proportions of regular updates in the presence of noise plummet below 10% for PSO, thus remarking the sensitivity of PSO to optimization problems subject to noise. Differently, those proportions in PSO-ER are over 30%, decreasing as the level of noise increases, yet remaining over 10%.

The proportions of blindness are smaller in PSO-ER (below 50%) than in PSO (over 50%) due to the greater accuracy to which the objective values of solutions in the former are estimated. The proportions of blindness in both algorithms are rather high considering that these are missed oppor- tunities to improve upon the best solutions found. The underlying reason to such high proportions of blindness is that particles are always prone to suffer from blindness at the next iteration regardless of what they do at the current iteration. Specifically, if a particle suffers from blindness or disorientation, or even if it has a regular operation at iteration t, the par- ticle is still prone to suffer from blindness to specific ranges of solutions at t + 1 (see Figure 3.2). The most common cause of blindness is blindness by memory, which is responsible for over 98% of the cases in PSO, and it ranges between 85–96% in PSO-ER increasing with the level of noise. Blindness by memory is vastly more common than blindness by the envi- ronment because the direction of the optimization problem is backwards,

3.4. RESULTS AND DISCUSSIONS 79 and hence the current solutions which are truly better than the personal best solutions will generally be more accurate as the proportional effect of noise will be smaller.

The proportions of disorientation in PSO-ER are larger (below 5%) than those in PSO (below 1%), but their presence in both algorithms is rather small compared to the proportions of regular operations and blindness. These proportions refer to those particles which mistakenly update their personal best solutions to worse solutions, thus causing a setback to the search process. The underlying reasons for such small proportions of dis- orientation are a) the direction of the optimization problem, b) the un- derestimation of objective values from the personal best solutions (pre- existing blindness), and c) the landscape of the objective search space af- fected by the noise therein. Firstly, the direction of the optimization prob- lem being backwards causes better solutions to be less corrupted by noise, and hence the overlapping of the probability density functions (represent- ing the noisy objective values) of adjacent solutions will be smaller. As such, the proportions of disorientation in backward optimization prob- lems will occur mostly at early iterations when the objective values of the solutions are generally bad and hence more likely to overlap (see Fig- ure 3.3). Secondly, pre-existing blindness creates an inverse correlation be- tween blindness and disorientation because larger underestimations of the objective values from the personal best solutions will require even larger underestimations of those from the current solutions; this, together with the backward direction of the optimization problem, will make disorienta- tion even less likely to happen (see Figure 3.4). Lastly, the landscape of the objective space and the noise therein will further determine the amount of overlapping between probability density functions of adjacent solutions in objective space. Regarding the causes of disorientation, the environment is responsible for over 95% of the cases of disorientation in PSO while it ranges between 83–93% in PSO-ER increasing with the level of noise. Disorientation by the environment is vastly more common than disorien-

0.0 0.1 0.2 0.3 f ~ (x t) f(x t) f ~ (y t−1) ∞

(a) Regular Operation

0.0 0.1 0.2 0.3 f ~ (x t) f(x t) f ~ (y t−1) ∞ (b) Disorientation 0.0 0.1 0.2 0.3 f(x t) f ~ (y t−1) f ~ (x t) ∞

(c) Blindness by the Environment

0.0 0.1 0.2 0.3 f ~ (y t−1) f ~ (x t) f(y t−1 ) ∞ (d) Blindness by Memory

Figure 3.2: Conditions leading to blindness. Figures 3.2a and 3.2b shows conditions of regular operation and disorientation that will blind a parti- cle at iteration t + 1. Figures 3.2c and 3.2d illustrate the causes of blindness for a particle at iteration t. The horizontal axis represents the objective values, and the vertical axis represents the probability density function of sampling noise. Gray areas cover the range of objective values to which a particle is (or will be) blinded. Solid and dashed lines represent the objec- tive values of the current and personal best solutions, respectively. Vertical lines assume the true objective value of the solution is estimated correctly, e.g. f (yt−1_{) = ˜f (y}t−1_).

3.4. RESULTS AND DISCUSSIONS 81 tation by memory because the direction of the optimization problem is backwards, and hence, when particles mistakenly update to worse solu- tions, their probability density functions will generally have larger stan- dard deviations than their once personal bests.

0 20 40 60 80 100 0 4 8 12 16 (a) Disorientation at σ06 0 20 40 60 80 100 0 4 8 12 16 (b) Disorientation at σ12 0 20 40 60 80 100 0 4 8 12 16 (c) Disorientation at σ18 0 20 40 60 80 100 0 4 8 12 16 (d) Disorientation at σ24 0 20 40 60 80 100 0 4 8 12 16 (e) Disorientation at σ30

Figure 3.3: Average number of disoriented particles in PSO-ER throughout

the iterations on F01 subject to different levels of noise. The horizontal

axis represents the iterations, and the vertical axis represents the average number of disoriented particles from all independent runs.