# An Improved MOEA/D with Optimal DE Schemes for Many-Objective Optimization Problems
## 3 MOEA/D with Optimal DE Schemes
### Framework of MOEA/D-oDE  
MOEA/D-oDE decomposes a many-objective problem into a set of single objective problems and solves them in parallel. Each decomposed single objective problem is handled with optimal DE schemes. The framework of MOEA/D-oDE is based on MOEA/D. We would like to point out that there are two distinguishing points from MOEA/D: (1) we select the WST decomposition approach to decompose the many-objective optimization problems; (2) optimal DE schemes instead of single DE operator schemes are employed in our experiments. Following the recounted methods, at each generation, MOEA/D-oDE with the WST approach maintains the following: (1) a population of \(N\) points \(x_{1},x_{2},\ldots,x_{N}\), where \(x_{i}\) is the present solution to the \(i\)-th subproblem; (2) reference ideal point \(z=(z_{1},z_{2}...,z_{m})^{T}\), where \(z_{j}\) is the best value for objective \(f_{j}\) that is found up to now, and \(\forall j=1,2,\ldots,m\); (3) the external population (EP), which is used to store the non-dominated solutions that have been found in the search process. The pseudocodes of our proposed algorithm MOEA/D-oDE are shown in Algorithm 1. As shown in Algorithm 1, in the initialization procedure of MOEA/D-oDE (Lines 8-17), we dispose of the neighborhood structure \(B(i)\) by computing the Euclidean distances between any two weight vectors and then arrange the \(T\) closest weight vectors with respect to each weight vector firstly. Furthermore, we generate the initial population, and for \(\forall i=1,\ldots,N\), evaluate \(F(x_{i})\) respectively, and mainly initialize \(z\) by the condition: \(z_{j}=\min\limits_{1\leq i\leq N}f_{j}(x_{i}),\forall j=1,\ldots,m\) in the end. In the update procedure of MOEA/D-oDE (Lines 19-27), if the stopping criteria are not satisfied, then go to Step 2.1 firstly, which does the reproduction by using optimal DE schemes, and then, the values that overstep the boundary should be repaired in Step 2.2. Secondly, we evaluate the function and update of \(z\) by the new minimum evaluation. Finally, through the \(g^{ust}(x^{\prime}|\lambda_{j},z)\leq g^{ust}(x_{j}|\lambda_{j},z)\), we renew the best solutions and obtain the final results. Figure 2: Flowchart of generating differential evolution (DE) operators. ```
1 Input:
2 MOP(1);
3\(N\): the number of the population size;
4\(\{\lambda_{1},\lambda_{2},\ldots,\lambda_{N}\}\): a set of \(N\) improved uniformly-distributed weight vectors;
5\(T\):the neighborhood size;
6 Output:
7 EP;
8
9 Step 1: Initialization
10 Step 1.1) Set \(EP=\emptyset\);
11 Step 1.2) Compute the Euclidean distances between any two weight vectors, and then dispose of the \(T\) closest weight vectors with respect to each weight vector. For each \(i=1,\ldots,N\), set \(B(i)=\{i_{1},i_{2},\ldots,i_{T}\}\), where \(\forall j\in B(i)\), \(\lambda_{j}\) are the \(T\) closest vectors to \(\lambda_{i}\);
12 Step 1.3) Generate the initial population randomly;
13 Step 1.4) \(\forall i=1,\ldots,N\), evaluate \(F(x_{i})\);
14 Step 1.5) Initialize \(z\): the original reference point \((z_{1},z_{2}...,z_{m})\) generated by the condition: \(z_{j}=\min\limits_{1\leq i\leq N}f_{j}(x_{i}),\forall j=1,\ldots,m\);
15 Step 2: Update
16 Step 2.1) Reproduction: Randomly select two indexes \(k\) and \(l\) from \(B(i)\), and generate a child solution \(x^{\prime}\) from the parents \(x_{k}\) and \(x_{l}\) by using the optimal DE schemes;
17 Step 2.2) Repair: Repair \(x^{\prime}\) by applying a problem-specific repair/improvement heuristic;
18 Step 2.3) Function evaluation: Evaluate \(F(x^{\prime})\);
19 Step 2.4) Update of \(z\): For each \(j=1,\ldots,m\), if \(z_{j}>f_{j}(x^{\prime})\), then set \(z_{j}=f_{j}(x^{\prime})\);
20 Step 2.5) Replacement/update of solutions: For each index \(j\in B(i)\), if \(g^{\text{next}}(x^{\prime}|\lambda_{j},z)\leq g^{\text{next}}(x_{j}|\lambda_{j },z)\) then set \(x_{j}=x^{\prime}\) and \(F(x_{j})=F(x^{\prime})\);
21 Step 2.6) Remove from EP all of the vectors dominated by \(F(x^{\prime})\).