# Memory-Enhanced Dynamic Multi-Objective Evolutionary Algorithm Based on \(L_{p}\) Decomposition
## 2 Related works  
### Problem Description and Basic Definitions  
Without loss of generality, an MOP with \(n\) decision variables and \(m\) objective functions can be described as:  
\[\left\{\begin{array}{l}\min_{x\in\Omega}F(x)=\left(f_{1}(x),f_{2}(x),...,f_ {m}(x)\right)^{\top},\\ \text{s. t. }g_{i}(x)\leq 0,i=1,2,...,s;h_{j}(x)=0,j=1,2,...,q,\end{array}\right. \tag{1}\]  
where \(x=(x_{1},x_{2},...,x_{n})^{T}\in\Omega\subset R^{n}\) is a vector of decision variables, \(\Omega\) is the \(n\)-dimensional decision space, and \(F(x)=(f_{1}f_{2},...,f_{m})^{T}\in\Lambda\subset R^{m}\) is an \(m\)-dimensional vector of \(m\) objective functions. \(R^{m}\) is called the objective space. The evaluation function \(F(x):\Omega\rightarrow\Lambda\) defines \(m\) mapping from the decision space to the objective space. \(g_{i}(x)\leq 0(i=1,2,...,s)\) are some inequality constraints, and \(h_{j}(x)=0(j=1,2,...,q)\) are some \(q\) equality constraints [24].  
The time variable of \(t\) is included in DMOP. Based on a static multi-objective problem, we can make an extension naturally. It involves an \(n\)-dimensional decision variable \(x\), \(m\)-objective functions, and the constraint conditions. The objective functions and the constraint conditions are functions of the decision variable \(x\). The mathematical model of a DMOP is expressed as:  
\[\left\{\begin{array}{l}\min_{x\in\Omega}F(x,t)=\left(f_{1}(x,t),f_{2}(x,t),...,f_{m}(x,t)\right)^{\top},\\ \text{s. t. }g_{i}(x,t)\leq 0,i=1,2,...,s;h_{j}(x,t)=0,j=1,2,...,q,\end{array}\right. \tag{2}\]  
where the evaluation function \(F(x,t)\), the inequality constraint function \(g_{i}(x,t)\), and the equality constraint function \(h_{j}(x,t)\) may change with time \(t\).  
**Definition 1** (The relation of Pareto dominance).: _For any two decision vectors \(u\) and \(v\), the vector \(u=(u_{1},u_{2},...,u_{m})^{T}\) dominates vector \(v=(v_{1},v_{2},...,v_{m})^{T}\), denoted by \(u\prec v\), only if \(\colon\forall k\in\{1,2,...,m\}\), \(\mathbf{u}_{k}\leq v_{k}\) and \(\exists l\in\{1,2,...,m\}\), \(u_{l}<v_{l}\)._  
**Definition 2** (Pareto optimal solution).: _The decision vector \(x\in\Omega\) is the Pareto optimal solution, if and only if \(\
eg\exists x^{\prime}\in\Omega\) satisfying \(F(x^{\prime},t)\prec F(x,t)\)._  
**Definition 3** (Pareto optimal set (POS(t))).: _The Pareto optimal set (POS) is the set that consists of all Pareto optimal solutions. The POS of a DMOP is defined as \(\text{POS}(t):\{x\in\Omega\,|\,\
eg\exists x^{\prime}\in\Omega,F(x^{\prime},t) \prec F(x,t)\,\}\)._  
**Definition 4** (Pareto optimal front (POF(t))).: _The Pareto optimal front (_**POF**_) of a DMOP is defined as \(\text{POF}(t):=\{F(x,t)\,|\,x\in\text{POS}(t)\,\). The Pareto optimal front (_**POF**_) is the set of all the Pareto optimal objective vectors._  
From the proposal of [PERSON] et al. [3], there are four different types of DMOPs according to the changes affecting the Pareto optimal front and the Pareto optimal set.  
* Type I: where **POS**(\(t\)) changes while **POF**(\(t\)) remains invariant.
* Type II: where both **POS**(\(t\)) and **POF**(\(t\)) change.
* Type III: where **POF**(\(t\)) changes while **POS**(\(t\)) remains invariant.
* Type IV: where both **POS**(\(t\)) and **POF**(\(t\)) remain invariant.  
This classification shows the difficulty of solving DMOPs by describing the combination of changes in the Pareto set and front.