5.3 Proposed Multiobjective Optimization Algorithms for Analog/RF Circuit Siz-
5.3.1 Hierarchical Nondominated Sorting Genetic Algorithm II (hNSGA-II) 120
5.3.1.5 Performance Analysis
• Parametric study : In this subsection, we perform a parametric study for population size parameter β of hierarchical polynomial mutation operator. Since the performance of proposed hierarchical scheme can be characterized for different population sizes of each local block (β), the performance can change with the change inβ. For a constant degreed, heighthcan be increased to postulate variations in both convergence and di- versity of Pareto optimal solutions at different values ofβfor a number of generations.
To measure the performance in terms of convergence towards optimal Pareto front and
diversity of solutions along Pareto front, three different quality metrics2 are evaluated, i.e.Generational Distance(GD),Inverted Generational Distance(IGD) andHypervol- ume (HV). GD metric [98] is used to measure minimum Euclidean distance between generated final individuals and sample points on true Pareto front. Smaller value of GD indicates a better convergence towards true Pareto front. On the other hand, both IGD metric [98] and HV metric [154] are a measure of both convergence and diversity of solutions along true Pareto front on a single scale. If the IGD value is close to zero, it means that the solutions have better convergence and diversity, and HV is a fair measure of the maximum area covered by the nondominated solutions with respect to a reference point.
In view of this, several experiments are carried out on ZDT1 test function for a number of generations and it can be observed from Figure 5.6 that with increase in β, solu- tion (optimal PF points) attempts to attain smaller (best) GD (Figure 5.15(a)) and IGD (Figure 5.15(b)) values and higher HV (Figure 5.15(c)) values at fewer generations.
One of the reasons for these best values at fewer generations for higherβ value can be due to the increase in number of offsprings in each local block with increase in β. It can be seen from Figure 5.15(c) that with the increase in number of generations, HV values also increase. However, the HV values remain unchanged after 500number of generations (forβ = 256 and700 generations for otherβ values). Since intermediate traces of PF exist at fewer generations, the population has to be evolved over sufficient number of generations (500 generations for β = 256) to account for the convergence and diversity metric to saturate during function evaluations. With more population, the diversity can be preserved by exploring extra search space by combining all new off- springs of each local block. Another reason for the best values at fewer generations can be the reduction in selection pressure by generating more offsprings through proposed hierarchical mutation strategy. With decrease in selection pressure, the convergence
2The details of quality metrics are described in Appendix C
5.3 Proposed Multiobjective Optimization Algorithms for Analog/RF Circuit Sizing
issues are addressed to a great extent by mitigating the instability in crowding distance method. However, as we know, β cannot be increased to any arbitrary value. Since β accounts for the variations in limiting distribution entropy of mutation operation, it should be kept within finite limits (0 < β < N) for better performance of hNSGA-II (as described in section 5.3.1.4).
0 50 100 300 500 700 1000
Number of generations 1
2 3 4 5 6
GD value (mean)
×10-3
β = 4 β = 16 β = 64 β = 256
(a)
50 100 300 500 700 900 1000
Number of generations 0
1 2 3 4
IGD value (mean)
×10-3
β = 4 β = 16 β = 64 β = 256
(b)
Number of generations
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Hypervolume (mean)
0.35 0.4 0.45
0.5 0.55
0.6 0.65
0.7
β = 4 β = 16 β = 64 β = 256
(c)
Figure 5.6: Performance of hNSGA-II on ZDT1 test function at different generations for various population sizes of local block (β), (a) mean GD values (b) mean IGD values (c) mean HV values.
• Comparison with Other Mutation Strategies : As mutation operator operates on an individual independent of other population members, it plays an important role in making overall search efficient [155]. Therefore, it is necessary to investigate the ef- fectiveness of proposed hierarchical mutation operator by comparing the performance with other mutation operators. Hierarchical scheme for six different mutation op- erators (Adaptive Levy mutation [156], Cauchy mutation [157, 158], uniform muta- tion [159], nonuniform mutation [159], Gaussian mutation [160] and Adaptive-mean
mutation [157]) are implemented and also incorporated to study the search efficiency of hNSGA-II framework. This study is extensive in evaluating the performance of proposed hierarchical polynomial mutation operator. Further, a total30decision vari- ables (p = 30) are considered during evaluation and each variable is bounded within xi ∈[0,1],where1≤ i ≤30. Crossover probability is kept at0.9and mutation prob- ability is set to1/pfor all mutation schemes. Both crossover and mutation distribution indices are set to 20during the entire process. Since various mutation operators are employed withinhNSGA-II framework, different parameters are addressed each time a mutation operator is exercised. During Cauchy mutation, a Cauchy density func- tion [158], fcauchy = π1t2+xt 2 is employed, where scale parameter t is set to 1during operation. The parameterαfixed at1.7for all experiments while using adaptive Levy mutation scheme. Box-Muller transform is adopted to generate Gaussian distribution from uniformly distributed numbers during Gaussian mutation operation. Population size of each local block,β is kept at256for achieving hierarchy during all operations.
All mutation operators exercised using hierarchical scheme are compared based on con- vergence and diversity of solutions (mean GD values, mean IGD values and mean HV values), while200population individuals are allowed to evolve over100generations on ZDT1 test function. For each mutation operation, 30independent runs are performed and the results are shown in Figure 5.7. It can be observed from Figure 5.7(a) and Fig- ure 5.7(b) that proposed hierarchical polynomial mutation operator performs better than other mutation operators as corresponding GD and IGD values are minimum. Further, it is observed from Figure 5.7(c) that the proposed hierarchical polynomial mutation operator has larger HV value as compared to other mutation operators. As the proposed hierarchical scheme is performed on top of polynomial mutation operator, the inherent probability distribution of creating an offspring population is similar to polynomial mu- tation operator, which is instrumental in the convergence of Pareto optimal solutions.
Moreover, the implicit generation of distinct fitness values in each generation favors the
5.3 Proposed Multiobjective Optimization Algorithms for Analog/RF Circuit Sizing
Cauchy Uniform NonuniformGaussian AdaptiveLevyAdaptiveMeanHierarchical
0 0.05
0.1 0.15 0.2 0.25 0.3
GD value
(a)
Cauchy Uniform NonuniformGaussian AdaptiveLevyAdaptiveMeanHierarchical
0 0.05
0.1 0.15 0.2 0.25 0.3
IGD value
(b)
Cauchy Uniform NonuniformGaussian AdaptiveLevyAdaptiveMeanHierarchical
0.3 0.4 0.5 0.6 0.7
HV value
(c)
Figure 5.7: Performance analysis of several mutation strategies on ZDT1 test function showcasing mean, (a) GD values (b) IGD values (c) HV values at different generations (100 to 1000).
preservation of diversity among Pareto optimal solutions.