# An Improved MOEA/D with Optimal DE Schemes for Many-Objective Optimization Problems
## 2 Preliminaries
### Differential Evolution's Trial Vectors  
DE is a kind of algorithm that deals with continuous optimization problems. To illustrate the basic concepts of DE, suppose that the objective function for this part is minimizing the \(f(\overrightarrow{x})\), \(\overrightarrow{x}=(x_{1},x_{2},\ldots,x_{D})\in\Re^{D}\), and the feasible solution region is \(\Omega=\prod\limits_{i=1}^{D}[L_{i},U_{i}]\). At generation \(G=0\), a random initial population \(\{\overrightarrow{x}_{i,0}=(x_{i,1,0},x_{i,2,0},\ldots,x_{i,D,0}),i=1,\ldots, N,j=1,\ldots,D\}\) is generated from the feasible solution region, where \(N\) is the population size. In the evolution of the DE algorithm, for each generation \(G\) of the current population, DE will create a mutation vector \(\overrightarrow{v}_{i,G}=(v_{i,1,G},v_{i,2,G},\ldots,v_{i,D,G})\) for each individual \(\overrightarrow{x}_{i,G}\), which is called a target vector. Five kinds of widely-used DE mutation operators are shown as follows.  
(1) \"DE/rand/1\" \[\overrightarrow{v}_{i,G}=\overrightarrow{x}_{r1,G}+F\cdot(\overrightarrow{x}_{ r2,G}-\overrightarrow{x}_{r3,G})\] (5) (2) \"DE/best/1\" \[\overrightarrow{v}_{i,G}=\overrightarrow{x}_{best,G}+F\cdot(\overrightarrow{x}_ {r1,G}-\overrightarrow{x}_{r2,G})\] (6) (3) \"DE/current-to-best/1\" \[\overrightarrow{v}_{i,G}=\overrightarrow{x}_{i,G}+F_{1}\cdot(\overrightarrow{x }_{best,G}-\overrightarrow{x}_{i,G})+F_{2}\cdot(\overrightarrow{x}_{r1,G}- \overrightarrow{x}_{r2,G})\] (7) (4) \"DE/best/2\" \[\overrightarrow{v}_{i,G}=\overrightarrow{x}_{best,G}+F_{1}\cdot(\overrightarrow{x }_{r1,G}-\overrightarrow{x}_{r2,G})+F_{2}\cdot(\overrightarrow{x}_{r3,G}- \overrightarrow{x}_{r4,G})\] (8)(5) \"DE/rand/2\"  
\[\overrightarrow{v}_{i,G}=\overrightarrow{x}_{r1,G}+F_{1}\cdot(\overrightarrow{x}_ {r2,G}-\overrightarrow{x}_{r3,G})+F_{2}\cdot(\overrightarrow{x}_{r4,G}- \overrightarrow{x}_{r5,G}) \tag{9}\]  
In the above five equations, \(r1,r2,r3,r4\) and \(r5\) are the independent individuals randomly selected from the population size and are different from \(i\). \(\overrightarrow{x}_{best,G}\) is the first-rate individual in the current population. Additionally, the parameters \(F\), \(F_{1}\) and \(F_{2}\) are called the scaling factors, and \(F\) and \(F_{2}\) are usually equal; all of them control the degree of mutation.  
After mutation, DE employs a crossover operator on \(\overrightarrow{v}_{i,G}\) and \(\overrightarrow{x}_{i,G}\) to generate a trial vector \(\overrightarrow{u}_{i,G}\) and \(\overrightarrow{u}_{i,G}=(u_{i,1,G},u_{i,2,G},\ldots,u_{i,D,G})\), described as follows:  
\[u_{i,j,G}=\left\{\begin{array}{ll}v_{i,j,G},&\text{if }rand_{j}(0,1)\leq Cr \text{ or }j=j_{rand}\\ x_{i,j,G},&\text{otherwise}\end{array}\right. \tag{10}\]  
where \(i=1,2,\ldots,N\), \(j=1,2,\ldots,D\), \(rand_{j}(0,1)\) is a random number uniformly distributed from zero to one, requiring to be generated for each \(j\). Additionally, \(Cr\in[0,1]\) is called the control parameter for crossover. The number \(j_{rand}\) is an integer in \([1,D]\), which is also chosen randomly. Using \(j_{rand}\), the trial vector \(\overrightarrow{u}_{i,G}\) could always be different from the target vector \(\overrightarrow{x}_{i,G}\). More details about DE and the DE's operator may be found in [22; 23; 30].