The results in Sections 3.2 and 3.3 reveal that the proposed features can improve the per- formance of MO optimization in terms of proximity, diversity and distribution under the influence of noise. In this section, the features of ELDP and GASS are applied to SPEA2 and NSGAII to examine if their effects can be reproduced in conventional MOEAs. The ELDP is used in place of the bit-flip mutation operator in SPEA2 and NSGAII, while the GASS
CHAPTER 3. 83 NSGAII 0 0.5 1 1.5 Generational Distance
NSGAII-RF SPEA2 SPEA2-RF
(a)
NSGAII NSGAII-RF SPEA2 SPEA2-RF 4
6 8
x 10-3 Spacing
(b)
NSGAII NSGAII-RF SPEA2 SPEA2-RF 0.7 0.8 0.9 1 Maximum Spread (c)
NSGAII NSGAII-RF SPEA2 SPEA2-RF 0
0.5 1
Hypervolume Ratio
(d)
Figure 3.40: Performance metric of (a) GD, (b) S, (c) MS, and (d) HVR for ZDT4 with 0% noise.
NSGAII NSGAII-RF SPEA2 SPEA2-RF 0
1 2
Generational Distance
(a)
NSGAII NSGAII-RF SPEA2 SPEA2-RF 0.05 0.1 0.15 0.2 0.25 Spacing (b)
NSGAII NSGAII-RF SPEA2 SPEA2-RF 0.6 0.7 0.8 0.9 1 Maximum Spread (c)
NSGAII NSGAII-RF SPEA2 SPEA2-RF 0 0.2 0.4 0.6 0.8 Hypervolume Ratio (d)
Figure 3.41: Performance metric of (a) GD, (b) S, (c) MS, and (d) HVR for ZDT4 with 20% noise.
is implemented in conjunction with existing selection schemes. The possibilistic archiving model is not implemented here, since the archiving strategy of SPEA2 and NSGAII plays an important role in defining the behavior of the algorithms.
It has been observed in previous section that SPEA2 and NSGAII can neither discover the global tradeoff for ZDT4 nor maintain a well-distributed set of individuals for FON. The performance of these two algorithms is also largely affected by noise in ZDT4 and FON. Hence, these two benchmark problems are used in the study here. NSGAII-RF and SPEA2- RF denotes the algorithm incorporated with the proposed features. The metric distribution of the simulation results for noiseless and noisy ZDT4 is shown in Figure 3.40(a)-(d) and Figure 3.41(a)-(d), respectively. Similarly, the performance of the algorithms for noiseless and noisy FON is shown in Figure 3.42(a)-(d) and Figure 3.43(a)-(d), respectively.
NSGAII NSGAII-RF SPEA2 SPEA2-RF 0 0.02 0.04 Generational Distance (a)
NSGAII NSGAII-RF SPEA2 SPEA2-RF 3 4 5 6 7 x 10-3 Spacing (b)
NSGAII NSGAII-RF SPEA2 SPEA2-RF 0.6 0.7 0.8 0.9 1 Maximum Spread (c)
NSGAII NSGAII-RF SPEA2 SPEA2-RF 0.75 0.8 0.85 0.9 0.95 Hypervolume Ratio (d)
Figure 3.42: Performance metric of (a) GD, (b) S, (c) MS, and (d) HVR for FON with 0% noise.
NSGAII NSGAII-RF SPEA2 SPEA2-RF 0
0.2 0.4
Generational Distance
(a)
NSGAII NSGAII-RF SPEA2 SPEA2-RF 0
0.2 0.4
Spacing
(b)
NSGAII NSGAII-RF SPEA2 SPEA2-RF 0 0.2 0.4 0.6 0.8 Maximum Spread (c)
NSGAII NSGAII-RF SPEA2 SPEA2-RF 0 0.2 0.4 0.6 Hypervolume Ratio (d)
Figure 3.43: Performance metric of (a) GD, (b) S, (c) MS, and (d) HVR for FON with 20% noise.
It can be observed from Figure 3.40 - Figure 3.43 that ELDP and GASS are capable of improving the performance of SPEA2 and NSGAII in terms of convergence and diversity of individuals along the tradeoff for ZDT4 and FON. In the case of ZDT4, the incorporation of the proposed features allows NSGAII-RF and SPEA2-RF to escape the local optima in ZDT4. In the case of FON, Figure 3.42 show that the incorporation of ELDP and GASS improves the performance in terms of GD, MS, and HVR. It can also be observed from Figure 3.43 that NSGAII-RF and SPEA2-RF have a slight edge over NSGAII and SPEA2 in almost all aspects of the MO optimization goals.
3.5
Conclusion
CHAPTER 3. 85
in this chapter, including an experiential learning directed perturbation operator that adapts the magnitude and direction of variation according to past experiences for fast convergence, a gene adaptation selection strategy that helps the evolutionary search in escaping from local optima, and a possibilistic archiving model based on the concept of possibility and necessity measures to deal with the problem of uncertainties. It has been shown in the comparative study that basic algorithm incorporating the proposed features exhibit com- petitive or superior performance in terms of proximity, diversity and distribution for both the noiseless and noisy benchmark problems. Besides, the working dynamics and parameter settings of ELDP and GASS with and without the presence of noise have been examined, which illustrate that the proposed features are robust to different parameter settings and the individual feature of ELDP and GASS plays an important role in the overall evolution- ary optimization process. Furthermore, it has been depicted that existing MOEAs such as SPEA2 and NSGAII incorporated with the proposed features of GASS and ELDP are capable of giving better convergence and population diversity along the global tradeoff for the benchmark problems with and without the presence of noise.
Hybrid Multi-objective
Evolutionary Design for Neural
Networks
In this chapter, we consider the design of artificial neural networks (ANNs) as an instance of noisy design problem. As mentioned by [107, 219], network architecture optimization is a noisy problem in which the same network structure can give rise to different fitness values due to different weight instantiations. Given that the intrinsic relationship between the architecture and the associated synaptic weights can be quite complex, the design method- ology would be flawed if we were to decouple these two properties during the training phase of the network. Local search is applied to optimize the synaptic weights with respect to any new ANN structure introduced to reduce the effects of noise.