An analysis of the effect of goal vector bounds

This section analyses the effect of goal vector bounds in PICEA-g. This analysis is conducted by considering six different sets of goal vectors generated using six different

gmax, see the left column of Figure 3.19 and Figure 3.20. In each case, PICEA-g is

executed independently for 31 runs (each run for 25 000 function evaluations) on the 2-objective WFG4 problem. Plots of Pareto front which has median hypervolume value are shown in the right column of Figure3.19and Figure3.20. In each figure, the shaded area represents the region dominated by a candidate solution. A and C represent the lower, gmin and upper, gmax bound, respectively. gmin is set to the coordinate origin

as the ideal point for WFG4-2 is (0,0). In respect to gmax, the six cases, respectively,

represent that

(i) the generated goal vectors only cover the knee (EF) of the Pareto optimal front: Figure3.19(a).

(ii) the generated goal vectors are all infeasible: Figure3.19(b).

(iii) more goal vectors are generated in the top-left part of the Pareto optimal front: Figure3.19(c).

(iv) more goal vectors are generated in the bottom-right part of the Pareto optimal front: Figure3.20(a).

(v) goal vectors are now generated within the bounds of ideal and nadir vector: Fig- ure3.20(b).

(vi) goal vectors are generated within the bounds of ideal and a relaxed form of the

(a) case i

(b) case ii

(c) case iii

Figure 3.19: Illustration of different goal vectors (Left) and the corresponding Pareto front (Right): Part I

(a) case iv

(b) case v

(c) case vi

Figure 3.20: Illustration of different goal vectors (Left) and the corresponding Pareto front (Right): Part II

The obtained non-dominated solutions for case (i) are shown in Figure 3.19(a). It is observed from the results that the obtained solutions are all constrained within the region of CEF. The reason is that candidate solutions (e.g. s2) inside the region of

ABCD are more likely to dominate more goal vectors and therefore gaining higher fitness than candidate solutions placed outside this region (e.g. s1 and s3). Hence, candidate solutions inside the region of ABCD are more likely to be propagated into the next generation. In addition, it is easy to understand that goal vectors in ABF ED are ineffective as these goal vectors cannot be dominated by any candidate solution, i.e., they have no contribution in offering comparability between alternative solutions; rather goal vectors should be generated in a feasible region so that there is a non-zero probability that at least one solution dominates the goal vectors.

The results for case (ii) are shown in Figure 3.19(b). Clearly, candidate solutions have not converged to the Pareto optimal front. The reason is that all goal vectors are infeasible. None of the candidate solutions is able to dominate any goal vector. Hence, candidate solutions are scored randomly and then PICEA-g behaviours in a similar way to random search.

Figure 3.19(c) shows the obtained solutions for case (iii). Solutions are biased towards the top-left of the Pareto optimal front as the generated goal vectors are biased to objective f1. Specifically, candidate solutions with smaller value of f1 (e.g., s1) would gain higher fitness by dominating more goal vectors than those with smaller value off2 (e.g., s3). Hence, solutions in the top-left are more likely to be propagated, leading a richer set of solutions in this region. In other words, more search effort is distributed on f1. Note that, case (iii) is just an example to illustrate that the search effort would be biased if goal vectors are not appropriately generated. Whether s1 would dominate more goal vectors or not is also conditioned on the problem geometry.

Case (iv) is similar to case (iii), see Figure 3.20(a). More goal vectors are generated in the bottom-right part. Therefore, solutions in the bottom-right region (e.g. s3) are more likely to gain higher fitness (compared to s1) and so are more likely to survive in the evolution.

Results for case (v) are shown in Figure3.20(b). Aand Crepresent theideal andnadir

vector, respectively. It is observed that solutions have converged well to the knee part of the Pareto optimal front while the extreme part of the Pareto optimal front is not sufficiently explored. Again, the reason is that compared with solutions in the knee region (e.g. s2), candidate solutions (e.g. s1 and s3) in the extreme parts can only dominate a few goal vectors and so have low fitness. That is to say, solutions in the extreme region are more likely to be disregarded; this degrades the solution diversity. Results for case (vi) are shown in Figure3.20(c). This time A is set to theideal point and C is set asα×nadir, α= 3 vector. From the results, we observe that the extreme solutions converge slightly better than the knee solutions. Additionally, goal vectors

generated in P GCQ are also ineffective as these goal vectors are dominated by all can- didate solutions, i.e. having no contribution in offering comparability between solutions; therefore we should not generate goal vectors in such region.

Based on the above analysis we conclude that different gmax have different influences

on the performance of PICEA-g. This enables us to adjust the search effort towards different objectives by using different gmax. In addition, from the above analysis it is

suggested that

• goal vectors that can be dominated by all candidate solutions or cannot be dom- inated by any candidate solution are ineffective, and should not be generated. Overall, the useful goal vectors are only those in the shaded region, see Figure3.21.

Figure 3.21: Illustration of the useful goal vectors.

• applying thenadir point asgmaxis not a good choice. In any case, thenadir point

is difficult to estimate. One reason has been mentioned in case (v), that is, the extreme part of the Pareto optimal front cannot be explored well. Another reason is that for some problems e.g., DTLZ1, the initial objective values are all outside the region enclosed by the nadir vector, even a relaxed form of nadir vector). In this case, PICEA-g would behave similarly to that of a random search, see Figure 3.19(c).