## 4 Materials and Methods
### DIRECT Algorithm for Multi-Objective Optimization
#### 4.2.1 Decomposition Based Strategy for Potentially Optimal Hyperrectangles Selecting  
As mentioned earlier, the selecting strategy for potentially optimal hyperrectangles is essential to the DIRECT algorithm, which means a possible modification or alternative that makes it suitable for multiple objectives is a straightforward manner for the DIRECT algorithm's extension. Decomposition is a basic idea which has been applied in many pop ular multi-objective algorithms, such as MOEA/D. Based on decomposition, the original problem is transformed into multiple scalar sub-problems in the direction of weight vectors, and then achieve optimization simultaneously. The penalty-based boundary intersection (PBI) approach is a commonly used decomposition method [16]. Through PBI, a MOP is converted into the following scalar optimization subproblem using the weight vector \(\lambda\) and the reference point \(z^{*}\) (see Figure 2).  
\[\text{minimize}\ \ g(x|\lambda,z^{*})=d_{1}+\theta d_{2}\] \[\text{where}\ \ \ d_{1}=\frac{||(z^{*}-F(x))^{T}\lambda||}{|| \lambda||}\] \[\text{and}\ \ d_{2}=||F(x)-(z^{*}-d_{1}\lambda)||. \tag{10}\]  
\(\theta>0\) is the pre-configured penalty parameter. Motivated by this idea, we propose a novel strategy to encourage to choose potentially optimal hyperrectangles in all directions.  
To make the strategy extend to multiple objectives, we first introduce a new indicator based on decomposition to quantify the quality of hyperrectangles (more specifically, the solutions at their centers). Suppose there are \(H\) weight vectors \(\lambda_{i},\ i\in[1,2,\ldots,H]\) and \(N\) hyperrectangles, the objective function value at the center of hyperrectangle \(j\) is \(f_{j},\ j\in[1,2,\ldots,N]\), and the reference point is denoted by \(z_{ref}\). Then, the procedure of generating the indicator is outlined in Algorithm 4. Following the idea of decomposition, the indicator maintains the ability to consider sub-problems in multiple reference directions at the same time.  
Figure 2: Illustration of PBI.  
As listed in Algorithm 4, the direction with largest \(d_{1}\) (maximum projection value) among all \(H\) sub-problems is used as the direction of the proposed indicator of every hyperrectangle, and then the largest \(d_{1}\) of each hyperrectangle will be normalized on the corresponding direction to obtain the value of the indicator (i.e., their contribution on the non-dominated front).  
On the basis of the proposed indicator, we can define a strategy for selecting potentially optimal hyperrectangles. That is, the hyperrectangle \(j\) is considered potentially optimal if there exists \(\hat{K}\) such that  
\[f_{j}^{s}-\hat{K}d_{j} \leq f_{i}^{s}-\hat{K}d_{i},\text{ for all }i=1,2,\ldots,m\] \[f_{j}^{s}-\hat{K}d_{j} \leq f_{min}^{s}-e|f_{min}^{s}| \tag{11}\]  
where \(e\) is a positive constant, \(m\) is the number of hyperrectangles to be compared, \(f_{j}^{s}\) is the indicator value at the center point of hyperrectangle \(j\), and \(d_{j}\) refers to the center-vertex distance of hyperrectangle \(j\), where \(f_{min}^{s}\) denotes the current optimal indicator value.  
With this unique selecting strategy, the DIRECT algorithm possesses the capacity to select potentially optimal hyperrectangles in all reference directions simultaneously, which will further improve the exploratory nature when the algorithm is stuck in a locally optimal Pareto set or a part of the global Pareto front.