Multi-Objective Optimisation Algorithms After proximity to the true Pareto-optimal front, diversity of solutions in the
2.5.1 Conflicts between Proximity and Diversity
The EMO process (and multi-objective optimisation process in general) is pre- sented with a multi-objective trade-off of its own. This trade-off arises due to the conflict between attaining ideal proximity and diversity in an approximation set. This is a bi-objective trade-off which exists in most cases where the true Pareto-optimal front is not known. In such a case it is not possible to determine whether the approximation set has converged to the true Pareto-optimal front, and therefore diversity preservation cannot become the focus of the remainder of the search. However, diversity preservation usually comes second to obtaining a good approximation set, as stated in [67]. The goal of diversity preservation is to preserve diversity along an approximation set as close to the Pareto-optimal front as possible.
The example in Figure 2.10 illustrates the trade-off between proximity and diversity. Set 2 has a more diverse population of solutions in comparison to Set 1; however Set 1 is closer in proximity to the Pareto-optimal front than Set 2. In this case, the better diversity offered by Set 2 is not as valuable as the proximity offered by Set 1.
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Figure 2.10: An illustration of the trade-off between proximity and diversity to the Pareto-optimal front of an objective function.
2.5.2
Methods of Diversity Preservation
There exist many novel concepts and variants of these concepts for the preserva- tion of diversity in a population throughout the EMO process. In this section a selection of methods of diversity preservation are described.
Adaptive Grid Algorithm
Bounded Pareto archiving (as in the Adaptive Grid Algorithm (AGA) strat- egy used in the PAES algorithm) is a simple yet powerful diversity preservation scheme which uses an adaptive grid to keep track of the density of solutions within divisions of the objective space [34]. To achieve this, a grid with a pre-set number of divisions is used to divide the objective space and when a solution is generated, its grid location is identified and associated with it. Each grid location is considered to contain its own sub-population, and information on how many
solutions in the archive are located within a certain grid location is available during the optimisation process, this has been illustrated in Figure 2.11.
0 0.2 0.4 0.6 0.8 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 Objective 1 Objective 2 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100
Figure 2.11: An example plot of a population and visualisation of grid divisions managed by an AGA.
When an archive has reached capacity and a new candidate solution is to be archived, the information tracked by the AGA is used to replace a solution in a grid location containing the highest number of solutions. When a candidate solution is non-dominated in regards to the current solution and the archive, the grid information is used to select the solution from the least populated grid location as the current (and parent) solution.
The AGA concept used in PAES (described in Section 2.7.2) later inspired several researchers, and was altered and deployed in multiple EMO algorithms such as the Pareto Envelope-based Selection Algorithm (PESA) (a population
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based version of PAES) [68], the Micro Genetic Algorithm [69], and the Domina- tion Based Multi-Objective Evolutionary Algorithm (-MOEA) [70].
Contributing Hypervolume
The contributing hypervolume indicator is an adaptation of the hypervolume indicator in order to be used as sorting criteria for selection operators, it has been used in the s-metric Selection Evolutionary Multi-Objective Algorithm (SMS- EMOA) [71] and the Hypervolume Estimation Algorithm for Multi-Objective Optimisation (HypE) [72]. The hypervolume indicator works by calculating the size of the objective space that has been dominated by an entire approximation set in regards to a specified reference point, where as the contributing hypervolume indicator assigns each solution in an approximation set with the size of the space covered by each solution exclusively. With this information the population can be sorted by the most dominant and diverse solutions. In addition, most contributing hypervolume indicator selection methods always assign solutions containing the extreme values for an objective with the highest hypervolume value. This has been illustrated in Figure 2.12 in two-dimensional space with a population of three solutions.
Although many state of the art EMO algorithms use the contributing hyper- volume as a sorting criteria for selection, its calculation becomes computationally infeasible as the number of problem objectives considered increase. Monte Carlo approximations have been used to speed up the calculation of the contributing hypervolume in [73], which through empirical experiments has shown that the method does not impair the quality of the approximation set. However, the speed increase provided by the Monte Carlo approximation method still results
f 1 f 2 X 1 X 2 X 3 fref
Figure 2.12: An example of the contributing hypervolume indicator in two- dimensional objective space.
in the contributing hypervolume indicator being infeasible on problems consisting of five objectives or more.
This particular measure of diversity preservation can also be used post-optimisation to reduce the size of a final approximation set produced by an optimiser, to a size that will not overwhelm and confuse a DM.
Crowding Comparison Operator
The crowded comparison operator is used in various stages of NSGA-II to guide its selection process towards an approximation set with uniformly spread out solutions. Associated with each individual in a population is two algorithm spe- cific properties: a non-domination rank, in which solutions are ranked by the number of solutions they are dominated by, found using the fast non-dominated
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sorting approach; and a local crowding distance, which is an estimation of the density of solutions surrounding a particular solution in the population [35, 74]. An illustration of this measure is given in Figure 2.13.
Figure 2.13: Calculation of the crowding-distance — points marked with solid markers are solutions of the same non-dominated rank.
Between two solutions with different non-domination ranks, the solution with the lower rank is given preference. However, if both solutions are of the same domination rank, then the solution which is located in a region with the least number of solutions is given preference.