# A Novel Multi-Objective Five-Elements Cycle Optimization Algorithm
## 2 Basic Concepts of MOO Problem and Related Research on MOEAs  
### Description of Multi-Objective Problems  
A Multiple-Objective Optimization(MOO) problem with \(m\) target functions and \(D\) decision variables (minimizing multi-objective problems) is represented by Equation (1) [21]:\[ \begin{cases}\min&Fun(X)&=(f_{1}(X),\ f_{2}(X),\ \ldots,\ f_{m}(X))^{T}\\ st.&g_{z}(X)&\geq\ 0,\ z=1,2,\ldots,\ Z\\ &h_{e}(X)&=\ 0\,,e=1,2,\ldots,\ E\end{cases}  \tag{1}\]  
where \(X=(x_{1},x_{2},\ \ldots\,x_{D})\) represents the \(D\)-dimensional control variable, \(Fun(X)\) stands for \(m\) contradictory objective functions; \(g_{z}(X)\geq 0\), \(z=1,2,\ \ldots,Z\) defines \(Z\) inequality constraints; \(h_{e}(X)=0\), \(e=1,2,\ \ldots\,E\) defines \(E\) equality constraint functions. The goal of multi-objective optimization is to find some certain \(X^{*}=(x_{1}^{*},x_{2}^{*},x_{3}^{*},\ \ldots\,x_{D}^{*})\), so that the objectives in \(Fun(X^{*})\) can be optimal at the same time under the constraint conditions. However, when solving MOO problems, since the objective functions may be contradictory, it is hardly possible to obtain one single optimal solution that enables every objective function to reach the optimal value, hence some compromise solutions should be made instead. These trade-off solutions enable each objective function to achieve a better condition simultaneously. This type of optimal solution set is called the Pareto solution set or the non-dominated solution set [22]. It was proposed by [PERSON] in 1896 and is named the Pareto optimal solution. The following basic definitions are often used when it comes to solving MOO problems with MOEAs [23]. **Definition 1**. : _Pareto optimal solution. Given a multi-objective optimization problem shown in Equation (1). If \(\forall\ X\in\Omega\) satisfies the following conditions then \(X^{*}\in\Omega\) is the optimal solution:_  
\[\wedge_{i\in r}(f_{i}(X)=f_{i}(X^{*})) \tag{2}\]  
_Or, there is at least one \(j\in\{1,2,\ \ldots\,m\}\), such that:_  
\[f_{j}(X)>f_{j}(X^{*}) \tag{3}\]  
_where \(r=(1,2,\ \ldots\,m)\), and \(\Omega\) satisfies the constraint in Equation (1), which is:_  
\[\Omega=\{X\in\mathbb{R}^{n}|g_{z}(X)\geq 0,h_{e}(X)=0,(z=1,2,\ \ldots\,Z;e=1,2,\ \ldots\,E)\} \tag{4}\]  
**Definition 2**. : _Dominance between individuals. Suppose \(X_{1}\) and \(X_{2}\) are two arbitrary, different individuals in the evolutionary population, if the following two conditions are met, we call \(X_{1}\) dominates \(X_{2}\):_  
_(1) For all sub-objectives, there are \(f_{r}(X_{1})\leq f_{r}(X_{2})\), \(r=(1,2,\ \ldots\,m)\)._  
_(2) There is at least one sub-objective that makes \(X_{1}\) better than \(X_{2}\), that is \(\exists j\in\{1,2,\ \ldots\,m\}\), such that \(f_{j}(X_{1})<f_{j}(X_{2})\)._  
_We call \(X_{1}\) non-dominated, or not inferior or dominant; \(X_{2}\) is dominated, expressed as \(X_{1}\succ X_{2}\), where \(\succ\) is the dominant relation._  
**Definition 3**. : _Pareto optimal front. Given a MOO problem shown in Equation (1) and its optimal solution set \(\{X^{*}\}\), the Pareto optimal front is defined as:_  
\[PF^{*}=\{Fun(X)=(f_{1}(X),f_{2}(X),\ \ldots\,f_{m}(X))|X\in X^{*}\} \tag{5}\]  
Generally, a set of trade-offs can be obtained for MOO problems, so the most important task in solving a MOO problem is to find a collection of Pareto optimal solutions as close to the true Pareto set as possible [24]. As shown in Figure 1, the solid line denotes the Pareto front of the two optimization objectives \(f_{1}\) and \(f_{2}\). It can be inferred that the Pareto front of a tri-objective problem constitutes a surface and the Pareto front of a many-objective optimization problem (with more than three function objectives) constitutes a hypersurface.