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This section discusses two difficulties that are usually encountered by decomposition based algorithms: problem geometry and many-objective optimisation.

4.3.1 Problem geometry

In decomposition based algorithms, each Pareto optimal solution corresponds to an optimal solution of a single objective problem that is defined by a weighted scalarising function. That is to say, once the weighted scalarising function is determined, the distribution of the obtained Pareto optimal solutions is determined. Furthermore, if the scalarising function is also chosen, the distribution of solutions would only be affected by the distribution of the employed weights.

Gu et al. (2012) andGiagkiozis et al.(2013) discussed what the optimal distribution of weights is for a specified Pareto front when using different scalarising functions. For ex- ample, when the Chevbyshev scalarising function is used, the optimal weight for search- ing for a solution x is (

1 f₁₍x) PM i=1_fi1(x) , 1 f₂₍x) PM i=1_fi1(x) ,· · ·, 1 fM(x) PM i=1_fi1(x)

). That is, given a weight vector w, the obtained Pareto optimal solution is along the search direction of _w1 (as long as the search direction _w1 does not point at a disconnected Pareto region).

Specifically, this relationship is described using Figure 4.6. The straight line from the reference point along the direction (_w1

i,1, 1

wi,2) can be described as f1wi,1 =f2wi,2. Note

that wi,j represents the jth component of weight vector wi. Line A intersects the

Pareto front at point s1. It is easy to see that min max(w1f(x)) = max(w1f(x1)), where x1 corresponds to the decision vector of s1. Thus, we can conclude that weight vector wi = (wi,1, wi,2) corresponds to a Pareto optimal solution along the direction of (_w1

i,1, 1

wi,2). In other words, the optimal weight vector corresponding to a Pareto

optimal solution x1 is determined by the vector (_f1(xf₁2₎₊(x_f1₂)_(x1),_f1(xf₁1₎₊(x_f1₂)_(x1)). It is worth mentioning that this conclusion is based on the assumption that there is a Pareto optimal

Figure 4.6: Illustration of the relation between weights and Pareto optimal solutions for a Chebyshev scalarising function.

solution along the defined search direction. If the search direction defined by a weight vector (e.g., w2) points at a disconnected region (termed boundary search direction), a solution at the boundary (e.g. s2) will be identified as the optimal solution for this weight vector instead. This is because the decision vectorx2 ofs2 produces the minimal value for the expression max(w2f(x)).

Given the above analysis, it is easy to see that the optimal distribution of weights for different problem geometries changes. Figure 4.7 illustrates the optimal distributions of weights for problems having linear, convex, concave or disconnected Pareto optimal fronts. Taking Figure 4.7(b) as an example, the optimal distribution of weights for a concave Pareto front is dense in the centre while sparse at the edge.

Overall, due to a lack of knowledge of the underlying problem geometry, it is usually not straightforward to determine a proper distribution of weightsa priori for decomposition based algorithms so as to obtain a set of evenly distributed solutions. Although the use of adaptive weights is potentially helpful in handling this issue (Jiang et al., 2011; Gu et al.,2012), it is suspected that adaptive weights might have a deleterious effect on an algorithm’s convergence performance (this will be discussed next).

4.3.2 Many-objective optimisation

Decomposition based algorithms using evenly distributed weights, such as MOEA/D, face difficulties on many-objective problems. This is because the number of Pareto optimal solutions that are required to describe the entire Pareto optimal front of a MaOP is very large (Ishibuchi et al., 2008b). In decomposition based algorithms each weight vector typically corresponds to one Pareto optimal solution. The evenly distributed weights are often initialised before the search and remain unchanged during the search. It

0 0.5 1 1.5 2 0 0.5 1 1.5 2

Pareto optimal front Optimal weights

(a) Linear Pareto front

0 0.5 1 1.5 2 0 0.5 1 1.5 2

Pareto optimal front Optimal weights

(b) Convex Pareto front

0 0.5 1 1.5 2 0 0.5 1 1.5 2

Pareto optimal front Optimal weights

(c) Concave Pareto front

0 0.5 1 1.5 2 0 0.5 1 1.5 2

Pareto optimal front Optimal weights

(d) Disconnected Pareto front

Figure 4.7: The optimal distributions of weights for different Pareto fronts in a 2- objective case using a Chebyshev scalarising function.

is therefore difficult to use a limited number of weights to obtain a full and representative approximation of the entire Pareto optimal front.

To illustrate this issue, we apply MOEA/D with 20 evenly distributed weights to solve the 2-objective DTLZ2 problem. Figure4.8(a)and Figure4.8(b)show the obtained non- dominated solutions in the last generation and in the archive respectively after running MOEA/D for 500 generations. It is obvious that the obtained solutions are not sufficient to cover the entire Pareto optimal front. It should be noted that due to the stochastic nature of MOEAs, neighbouring solutions of thesw are likely to be obtained during the

search. sw is referred as the optimal solution of a single objective problem defined by

the weighted scalarising functiong(x|w). However, it is less likely to find solutions that are distant fromsw, see Figure4.8(b).

A natural way to address this limitation, i.e., a lack of solution diversity, is by employing a large number of weights. However, it is argued that compared with the number of solutions required to describe the entire Pareto optimal front, the number of employed

0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 Non−dominated solutions Optimal weights

(a) Solutions in the last generation

0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 Offline solutions Optimal weights

(b) Solutions in the offline archive

Figure 4.8: An approximation of Pareto front of 2-objective DTLZ2 obtained by MOEA/D using 20 weights.

weights is always relatively small. Besides, for some decomposition based MOEAs, e.g., MOEA/D, the population size is required to be equal to the number of weights. It is not easy to strike an effective balance between the population size and the generations with a fixed computational budget – the larger the population size, the the more the beneficial dynamics of evolution are curtailed.

(a) fixed weights (b) non-fixed weights

Figure 4.9: Illustration of the search behaviour using fixed weights and non-fixed weights.

Another alternative is to use non-fixed weights. Typically, non-fixed weights could be either randomly generated or adaptively modified during the search. The use of non- fixed weights enables MOEAs to have more opportunities to explore different regions, thereby obtaining a set of diversified solutions. However, this might slow down the

convergence speed of an algorithm. When using fixed weights, solutions are guided towards the Pareto optimal front along the search directions constructed by the weights, see Figure4.9(a). When using non-fixed weights, the constructed search directions keep changing. This suggests that solutions are guided towards the Pareto optimal front in a polyline trajectory as shown in Figure4.9(b), that is, the convergence speed is degraded. Certainly, it should be admitted that in some case, e.g., multi-modal problems, the use of random/adaptive weights is helpful to maintain diversified solutions and to prevent the algorithm being trapped in the local optima, resulting in better convergence. Overall, decomposition based algorithms face difficulties with many-objective optimisa- tion. This issue has not received much attention. To the best of the author’s knowledge, none of the existing decomposition based algorithms that use adaptive weights, such as Hughes (2007), Li and Landa-Silva (2011) andGu et al. (2012) can effectively strike a balance between exploitation (convergence) and exploration (diversity). Therefore, it is proposed to develop an effective weights adaptation strategy to address this issue.