## 3 The Proposed NSPSO Sampling Algorithm
### Normal Search Pattern  
Most existing optimization algorithms for UMOPs or MMOPs tends to guide particles to certain locations according to the global and local optimal locations (Figure 3(left)). Such a feature helps UMOPs or MMOPs to let particles converge to one or several locations [25,26]. However, for ROMPs such as the sampling process of LSF described above, continuity of the solutions will be ignored using such a searching pattern, leading to a poor approximation of the LSF. To deal with this issue, normal search pattern was designed. Take a 2-D problem as an example: the two blue particles in Figure 3(right) represent the best locations of a niche. Nearby particles were perpendicularly guided towards their connecting line so that the continuous feature and local geometric characteristics can be considered. Generally, particles were guided to the normal direction of the hyperplane formed by several best locations of a niche. For a \(d\)-dimensional RMOP, normal search pattern record \(n\) distance-based neighbor [27] for each particle and select the best \(d\) particles to form a \(d-1\)-dimensional hyperplane. By using the normal search pattern, continuity of the solution can be fully utilized.  
However, this also raises an instability problem as shown in Figure 4. Particle A was guided towards the connecting line of two neighbor optimal locations. However, the projection point on the line is not an optimal position. Therefore, the particle may vibrate between the optimal position and the projection point. This issue is serious when the optimal region is highly nonlinear, or the number of particles is small.  
Figure 3: Point search (**left**) and normal search (**right**).  
To address this issue, we added a compensation term to the normal search pattern (the second term in Equation (8)). To sum up, the proposed normal search pattern can be defined as Equations (8)-(10).  
\[\mathbf{N}_{i}(t)=\underbrace{n_{i}(t)}_{1}-\delta^{-(f(\mathbf{x}_{i}(t))-f(\mathbf{p}_{s} (t)))}_{1}\cdot\mathbf{n}_{\mathbf{p}_{i}}(t) \tag{8}\]  
\[\mathbf{n}_{i}(t)=\mathbf{\alpha}_{i,Q_{i}^{\text{ms}}}^{\text{o}}(t)\cdot d_{Q_{i}^{ \text{ms}}}^{\text{i}}(t) \tag{9}\]  
\[\mathbf{n}_{\mathbf{p}_{i}}(t)=\mathbf{\alpha}_{i,Q_{i}^{\text{ms}}}^{\text{o}}(t)\cdot d_{Q _{i}^{\text{ms}}}^{\text{i}}(t) \tag{10}\]  
where \(N_{i}(t)\) is the normal search vector that will be exerted on particle \(i\) at iteration \(t\); \(\mathbf{n}_{i}(t)\) is the uncorrected normal search vector; \(\mathbf{\alpha}_{i,Q_{i}^{\text{ms}}}^{\text{o}}(t)\) and \(d_{Q_{i}^{\text{ms}}}^{\text{i}}(t)\) are the unit vector and minimum distance between particle \(i\) to the hyperplane \(Q_{i}^{\text{ms}}\) formed by the best locations of the neighbors of particle \(i\), respectively; \(n\)s is the niche population size; \(d\) denotes the problem dimension. The second term is the compensation term that was intended to offset \(\mathbf{n}_{i}(t)\) when particle \(i\) is near the optimal location. \(\mathbf{p}_{i}\) is the personal best location of particle \(i\) so far; \(\delta>1\) is a parameter that controls the strength of the compensation term; \(d_{Q_{i}^{\text{ms}}}^{\text{p}_{i}}(t)\) is the minimum distance between \(\mathbf{p}_{i}\) and \(Q_{i}^{\text{ms}}\).  
As can be seen from Equation (8), implementation of the normal search can be divided into two parts. The first term is the searching vector of particles along the normal direction of the corresponding hyperplane. The second term is the normal search correction item based on the adaptive activation strategy. Its freezing coefficient \(\delta^{-(f(\mathbf{x}_{i}(t))-f(\mathbf{p}_{s}(t)))}\) is used to determine whether the normal search mode is activated or not. Apparently the second term is effective only when the current particle is near optimal, so that the optimal region can capture this particle to deal with the problem as shown in Figure 4 properly.