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It is experimentally analyzedwhether the1-greedy SMS-EMOA can robustly obtain the hypervolume-optimal distribution of points for the approximation of piecewise continuous Pareto fronts with different curvature (convex to concave). We consider the local 1-greediness as defined in Section 3.3.1. Recall that a local 1-greedy solvable problem is also 1-greedy solvable. We perform a comprehensive study on the set of simple test functions T1-T5 (Sec. 3.3.1), whereas we restrict our analyses to small populations, due to our results in Section 3.3.2.

Pre-experimental planning To gain a deeper understanding of the hypervolume progress, we first analyze hypervolume contributions and local 1-greedy steps. Hypervolume contributions: Intuitively, one may assume that the hypervolume contributions of individuals tend to equal values for all points of an optimally distributed set since, otherwise, a solution can move closer to the point with a higher contribution. This assumption is wrong, as demonstrated for T4 with α _∈ {1/3,1/2,1,2,3_}resulting in two concave fronts forα <1, convex fronts forα >1, and a linear front for α = 1. The analytically determined optimal positions of the points w. r. t. the reference point r = (1,1)> are shown in Fig. 3.10 (left) and the corresponding hypervolume contributions on the right side. The graphs are symmetric to the bisecting line. Accordingly, the distributions are symmetric to the

0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.2 0.4 0.6 0.8 1 f2 f1 α=1/3 α=1/2 α=1 α=2 α=3 0 0.01 0.02 0.03 0.04 0.05 1 2 3 4 5 contribution point’s index α=1/3 α=1/2 α₌₁ α₌₂ α₌₃

Fig. 3.10: Optimal positions of points on Pareto fronts of T4 with different curvature (left)

and corresponding hypervolume contributions (right), w. r. t. the reference point r= (1,1)>. The distributions are symmetric and the contributions are not all equal.

Fig. 3.11: Left: Areas of acceptance/rejection of points due to their hypervolume contri-

bution, relative to the decision space variables of a population on the Pareto front of T1. Improving areas are adjacent to the points.

Right: Progress of moving the points from their starting positions to their optimal positions by a local search(5 + 1)-SMS-EMOA on T4 (α= 1/3).

middle point on the bisecting line, and so are the hypervolume contributions. The contributions are displayed sorted according to the first objective. For α < 3, the values tend to grow with increasingα. On concave Pareto fronts, the middle point has the highest contribution and values decrease monotonic towards the boundaries, whereas it is the other way round for convex Pareto fronts, so that the knee point has the lowest contribution. Only on the linear front, all contributions are equal, and the points are equally spaced as proved in Lemma 2.7.

Local 1-greedyness: Fig. 3.11 (left) shows the areas of acceptance and rejection of points, exemplarily for a randomly initialized population on T1 (analogous results have been observed for T2-T5). Note that the acceptance regions are directly

adjacent to the current points, so that a point can be improved by a local movement. To further investigate the effect of single local refinements, a local search SMS- EMOA is applied, with variation only by Gaussian mutation with the small step sizeσ = 0.01. Fig. 3.11(right) displays the progressof the decision space variables of this local search (5 + 1)-SMS-EMOA on T4 with α = 1/3 w. r. t. the fixed reference point r0 = (2,2)>. As starting positions, the optimal distribution for the closer reference point r = (1,1)> is used. The algorithm is able to guide the solutions from the old to the new optimal positions. [. . . ] For the new reference point r0 the optimal points move closer to the boundaries of the Pareto front as there is more hypervolume to gain due to the farther reference point. The other points follow these extremal solutions to cover the resulting distance.

The following experiment shall support the arising assumption that even a local search-based (µ+1) SMS-EMOA converges towards the optimal population.

Task Check the hypothesis that the local search SMS-EMOA is able to approxi- mate the optimally distributed subset of the Pareto front of the given continuous test problems T1-T5 for fixed population sizes µ _{∈ {}1, . . . ,6_} with an accuracy limited by the step sizeσ = 0.01.

Setup For each test function T1-T5 and population sizes µ _{∈ {}1, . . . ,6_}, ap- proximations for the hypervolume-optimal distribution are globally calculated by the MATLAB implementation of the (5,10)-CMA-ES by Hansen and Ostermeier (2001), where no limit on the function evaluations, but a lower limit on σi (i =

1, . . . , µ) of 10−12 _{is specified. We trust in the quality of the results due to the}

experiences in Sec. 3.3.2. For each configuration 10,000 runs of the local search SMS-EMOA are performed using different random initializations and the results afterµ_·1,000 generations are compared to the approximations found by the global optimization of the CMA-ES. A run is denoted as failed when the hypervolume of the found approximation is below99% of the approximated optimal one.

Results/Observations The local search SMS-EMOA detects the optimal distri- bution in all runs for the convex and concave testfunctions T1-T5 except for 10%

of the initializations on the concave-convex Pareto front T5 for µ= 1. When the initial solution is situated close to the left border (x <0.1), the local search SMS- EMOA converges to the left border, which indicates the optimum for the concave part of the Pareto front, instead of detecting the globally optimal position in the inflection point.

Discussion Based on thorough experimentation, it can be assumed that even a local search SMS-EMOA robustly detects the globally optimal distribution in cases where the sign of the second derivative with respect to the first objective does not change. However, due to emerging effects of the local refinements and

their interaction, for higher population sizes, this result seems to hold also for concave-convex Pareto fronts.

3.3.4 Conclusions

We investigated the process of the SMS-EMOA when optimizing the distribution of points on the Pareto front in order to maximize the hypervolume. Our posed question was, whether the steady-state (or 1-greedy) selection of exchanging only one individual per generation is able to reach a global optimum of the population’s hypervolume which is defined w. r. t. a fixed reference point.

To deeper understand the optimization process, we analyzed a counter-example. It is revealed that the local optimum where the EMOA gets stuck is not a singularity but has an actual strong attractor so that also µ-greedy or non-elitist algorithms fail to detect the global optimum. This result is a step forward from theoretical possibilities to more practical probabilities. The constructed counter examples appear artificial as the Pareto fronts are disconnected and one may think that the non-optimality is achieved by forbidding points at the right positions. However, connecting the parts shall keep the negative properties while making the function continuously differentiable, so we conjecture that counter examples also exist for this class of problems where an arbitrary movement of points is possible.

We proved that the hypervolume is a concave function in case of a linear Pareto front, thus the function does not have a non-global local optimum and the contin- uous improvement of the hypervolume results in the global optimum. Thorough experiments showed that the optimal hypervolume is reliably detected by the 1- greedy SMS-EMOA on the chosen examples of continuously differentiable functions with connected Pareto fronts. These results strongly suggests that these functions are 1-greedy solvable, i.e., there is no configuration from where the optimum cannot be reached. Failures have only been detected for a (1+1)-SMS-EMOA, so we think the risk of not being 1-greedy solvable decreases with increasing population size of the EMOA.

Recall that we considered a model scenario with a fixed reference point, so the results do not directly apply to the SMS-EMOA with the normally used adaptive reference point. For a large population size, the scenarios shall behave similar as the reference point is expected to change only slightly after the points are spread along the whole Pareto front. For these cases, we experimentally did not observe any failure on the considered functions. For small population sizes, we witnessed some failures of getting stuck in a local optimum. These problems shall not occur with an adaptive reference point, where the optimization problem is dynamic due to the changing reference point, so that the population is unlikely to get stuck. This again advocates the concept of the adaptive reference point.

This topic is still under examination as the recent work by Bringmann and Friedrich (2011) shows. Our conjectures are still neither proved nor disproved.

Open problems are to identify other function classes where the 1-greedy concept suffices or if not—to give the lowestk so that a k-greedy selection succeeds. Even more desirable than analyzing the general possibility of solvability (if a path to the optimum exists from each configuration) is the consideration of the probabilistic solvability, i.e., analyzing how likely it is to reach the global optimum. Our exper- imental studies are a first step, whereas more insights are worthwhile to generalize our results.