# Multi-Objective Optimization of Square Corrugation Multilayer Nested Structures
## 3 The Methodology of Multi-Objective Optimization
### The DCNSGA-III Algorithm  
According to [PERSON] et al. (2018), the DCNSGA-III method is proposed, which is created by employing reference-point-based NSGA-III to resolve constrained multi-objective optimization problems (CMaOPs). The NSGA-III method is advantageous to handle constraints and optimize objectives concurrently, which can make a CMaOP dynamic ([PERSON] et al., 2018). The DCNSGA-III method performs great competitiveness compared with other sophisticated CMaOEAs and is competent for solving CMaOPs. The general process of DCNSGA-III is described as follows.  
Generate CMaOP framework.  
The DCNSGA-III, which integrates widely used NSGA-III and a problem conversion means, is applied to resolve CMaOP. Algorithm 1 displays the trunk of how the DCNSGA-III solves the CMaOP.  
```
1 Search incipient population \(P_{0}\);
2 Form the incipient relaxed constrained margin \(\delta=\delta^{(0)}\), and make the iteration state \(t=0\);
3 whilethe termination requirement is not satisfieddo
4 Shrink the boundary: \(\delta=\delta^{(t+1)}\);
5 Renew population \(\delta\)-feasibility, \(t=t+1\);
6 Obtain the offspring population \(O_{t}\) in terms of Algorithm 2;
7 Renew the parent population \(P_{t+1}\) in terms of Algorithm 3;
8 end while
9 return non-dominated solutions.
```  
**Algorithm 1**The DCNSGA-III to solve the framework of CMaOP  
At the beginning of the framework, the population \(P_{0}\) is generated randomly by DCNSGA-III from unitary search space \(X\). The non-constraint degree of every solution is computed subsequently, and the maximal non-constraint degree is regarded as the incipient constraint bounds \(\delta\). Thus, any solution in population \(P_{0}\) is \(\delta\)-feasible. In subsequent iterations, the constraint bounds will decrease over the evolution of the population. In addition, a modified mating and environment selection mechanism is utilized to obtain highly qualified \(\delta\)-feasible solutions.  
Select mating parents.  
It is inevitable that \(\delta\)-infeasible solutions are generated. In this connection, a binary tournament selection is employed to generate the offspring operator, which picks two individuals from \(P_{t}\). To be specific, the tournament selection intends to select an \(\delta\)-feasible solution instead of an \(\delta\)-infeasible solution or an \(\delta\)-infeasible with a minor non-constraint level instead of an \(\delta\)-infeasible with a major non-constraint level so that more \(\delta\)-feasible offspring solutions are obtained. Algorithm 2 describes the main procedure of tournament selection.  
```
Input: alternative solutions \(x_{1}\) and \(x_{2}\) Output: breeding parent \(p\)
1 if\(x_{1}\) is \(\delta\)-feasible solution and \(x_{2}\) is \(\delta\)-infeasible then
2\(p\gets x_{1}\);
3 else if\(x_{1}\) is \(\delta\)-infeasible solution and \(x_{2}\) is \(\delta\)-feasible then
4\(p\gets x_{2}\);
5 else if\(x_{1}\) and \(x_{2}\) all are \(\delta\)-infeasible then
6 if\(cv\)\((x_{1})\) < \(cv\)\((x_{1})\)then
7\(p\gets x_{1}\);
8 else if\(cv\)\((x_{1})\) > \(cv\)\((x_{1})\)then
9\(p\gets x_{2}\);
10 else
11\(p\leftarrow\) random \((x_{1},x_{2})\);
12 end if
13 else
14\(p\leftarrow\) random \((x_{1},x_{2})\);
15 end if
16 return \(p\).
```  
**Algorithm 2**The procedure of tournament selection