1e+02 1e+03 1e+04 1e+05 1e+06 1e+07
25 50 100 200 400
ZEROMAX - ONEMAX
1e+02 1e+03 1e+04 1e+05 1e+06 1e+07
25 50 100 200 400
TRAP - INVERSE TRAP
MO-GOMEA k=5 MO-GOMEA k = 1 -> +m
MO-GOMEA k = m+1 -> +(m+1) MO-GOMEA k = m+1 -> +1
Figure 3.12: Experiments on eliminating the population size parameter for MO-GOMEA. Hori-zontal axis: Problem size. Vertical axis: The number of evaluations until the whole Pareto-optimal front is obtained.
results on both test problems and also the most similar performance with the base population-sizing-free MO-GOMEA k = 5. Therefore, we decide that the third k-free scheme m + 1 → +1 is the most suitable method (among the methods tested here) to remove the number of clusters parameter.
3.7. Performance of the MO-GOMEA and the influ-ence of mutation operators
In this section, we conduct experiments on all benchmark problems (as presented in Section 3.3) to study the performance of our newly created parameter setting-free MO-GOMEA and the influence of the use of different mutation operators.
3.7.1. Design of mutation operators
The standard version of MO-GOMEA works well without any mutation operators.
However, empirical results showed that mutation could be beneficial to the per-formance of MO-GOMEA on some problems. We here implement mutation as an additional component that the user can easily switch on or off as desired. Muta-tion is employed with some probability pmon every problem variable in the linkage set at every mixing step during the Optimal Mixing procedure. Mutation should be performed after copying values from a donor to the current solution and before fitness evaluation of the intermediate solution. We propose two mutation opera-tors: weak mutation and strong mutation. Weak mutation uses a fixed mutation probability pm= 1/l (l is the number of problem variables). Strong mutation uses an adaptive mutation probability pm = 1/lFi, where lFi =
Fi
is the number of
problem variables in a linkage set Fi.
3.7.2. Scalable benchmark problems
1e+02 1e+03 1e+04 1e+05 1e+06 1e+07
25 50 100 200 400
ZEROMAX - ONEMAX
1e+02 1e+03 1e+04 1e+05 1e+06 1e+07
25 50 100 200 400
TRAP - INVERSE TRAP
1e+02 1e+03 1e+04 1e+05 1e+06 1e+07
25 50 100 200 400
LOTZ
MO-GOMEA no mut. MO-GOMEA weak mut. MO-GOMEA strong mut.
Figure 3.13: Performance of MO-GOMEA without mutation and with weak/strong mutation on scalable benchmark problems. Horizontal axis: Problem size. Vertical axis: The number of evalu-ations until the whole Pareto-optimal front is obtained.
Figure 3.13 shows that MO-GOMEA with the strong mutation operator is the fastest solver for Zeromax-Onemax. As all problem variables are independent from each other, linkage learning is not required to solve Zeromax-Onemax and a muta-tion operator with high mutamuta-tion probability pmwill help obtain the whole Pareto-optimal front quicker by simply performing bit-flips to try out different alternatives without disrupting any building blocks. Such aggressive mutation operators, how-ever, worsen the performance of MO-GOMEA on problems that require linkage learning such as Trap-Inverse Trap. Weak mutation has little influence on the scal-ability of MO-GOMEA in solving Zeromax-Onemax and Trap-Inverse Trap because its mutation probability becomes increasingly smaller as the problem size increases.
3.7. Performance of the MO-GOMEA and the influence of mutation operators
While MO-GOMEA with weak mutation performs slightly better than MO-GOMEA without mutation in solving Zeromax-Onemax, there is no statistically significant difference between the two optimizers in solving Trap-Inverse Trap.
MO-GOMEA without mutation operators has difficulties in solving the LOTZ problem, especially in approaching two extreme solutions (i.e., a string of all 1s and a string of all 0s). As discussed in Section 3.5.2, only leading 1 bits and trailing 0 bits contribute to the objective values during the optimization process. It can be seen that all solutions containing trailing 1 bits and leading 0 bits can easily be dominated and replaced by any solutions ending with a 0 and beginning with a 1. Those trailing 1s and leading 0s, however, are essential to the construction of the extreme solutions at the later stage of the optimization process when they meet the leading 1s and trailing 0s. Gene-pool Optimal Mixing, therefore, cannot find any donor with trailing 1s nor leading 0s remaining in the populations when neces-sary. This problem, however, can be alleviated by employing mutation operators.
Both weak and strong mutation operators significantly improve the performance of MO-GOMEA when solving the LOTZ problem by bringing back the prematurely disappeared bit values.
While these three benchmarks are convenient for scalability benchmarking be-cause all Pareto-optimal solutions can be computed analytically, their Pareto-optimal fronts always have the shape of a straight line and they therefore do not resem-ble real-world optimization proresem-blems. In the following section, we will consider MAXCUT, a problem whose Pareto-optimal fronts can have multiple regions with different shapes, and knapsack, an optimization problem with constraints.
3.7.3. MAXCUT & Knapsack
Figure 3.14 shows the average convergence behavior of MO-GOMEA and the in-fluence of weak and strong mutation in solving four MAXCUT problem instances.
The weak mutation operator has little influence on the performance of MO-GOMEA.
The strong mutation operator improves the convergence in the 100-vertex instance but worsens the performance in the 50-vertex instance; the differences are found to be statistically significant. Nevertheless, all the performance gaps between MO-GOMEA without mutation and MO-MO-GOMEA with mutation are small. In these MAXCUT problem instances, the linkage information-guided solution recombina-tion of Gene-pool Optimal Mixing is effective enough for MO-GOMEA to explore the search space without requiring any mutation operators.
Figure 3.15 shows the average convergence performance of MO-GOMEA and the influence of mutation operators in solving four multi-objective Knapsack problem instances. MO-GOMEA without mutation has the best convergence performance in all cases. The weak mutation operator exhibits little effects while the strong mutation operator severely degrades the performance of MO-GOMEA. Mutation operators with high mutation rates pm can disrupt important building blocks ac-quired by the linkage learning and Gene-pool Optimal Mixing procedures.
While mutation is a crucial variation operator of other MOEAs, it is not a required component for MO-GOMEA. Among all benchmark results, mutation op-erators only significantly deliver positive improvements when solving the LOTZ
problem. Therefore, we conclude that the type of mutation and the probability of mutation should be, much like local search operators, considered as highly problem-specific and, therefore, should not be seen as a default component of MO-GOMEA.
0
0e+00 1e+04 2e+04 3e+04 4e+04 5e+04
MAXCUT 12
0e+00 1e+05 2e+05 3e+05 4e+05 5e+05
MAXCUT 25
0e+00 1e+06 2e+06 3e+06 4e+06 5e+06
MAXCUT 50
0e+00 4e+06 8e+06 1e+07 2e+07 2e+07
MAXCUT 100
MO-GOMEA no mut. MO-GOMEA weak mut. MO-GOMEA strong mut.
Figure 3.14: Average convergence performance of MO-GOMEA without mutation and with weak/strong mutation on MAXCUT. Horizontal axis: number of evaluations (both objectives per evaluation). Vertical axis: DPF→S.