# Memory-Enhanced Dynamic Multi-Objective Evolutionary Algorithm Based on \(L_{p}\) Decomposition
## 2 Related works
### Decomposition Methods  
Decomposition-based multi-objective evolutionary algorithms have become an extremely prevailing framework for multi-objective optimization. Their main idea is to decompose an MOP into a number of scalaring sub-problems and solve them in parallel, where the decomposition method plays an extremely important role. There are three commonly used decomposition methods [23]: weighted sum approach (WS) [25], Tchebycheff approach (TCH) [25] and the penalty-based boundary intersection approach (PBI) [23; 26], described as follows.  
* Weighted Sum (WS) approach  
The DMOPs of Formula (2) are decomposed into \(N\) dynamic scalar optimization subproblems by using \(N\) different uniformly distributed weight vectors \(\lambda^{i}=(\lambda^{i}_{1},\lambda^{i}_{2},...,\lambda^{i}_{m})(i=1,2,...,N)\), where for all \(i=1,2,..,N\), \(\lambda^{i}_{j}\geq 0\) and \(\sum\limits_{j=1}^{m}\lambda^{i}_{j}=1\).  
With this approach, the decomposed scalar subproblem can be described as follows:\[\left\{\begin{array}{l}\min\limits_{x\in\Omega}g^{\text{zvs}}(x,t\left|\lambda \right.)=\sum\limits_{i=1}^{m}\lambda_{i}f_{i}(x,t),\\ \text{s. t. }g_{i}(x,t)\leq 0,i=1,2,...,s;h_{j}(x,t)=0,j=1,2,...,q.\end{array}\right. \tag{3}\]  
\(x=(x_{1},x_{2},...,x_{n})^{T}\) are \(n\) variables to be optimized. We can use different weight vectors to generate a set of different scalar optimization problems. \(g_{i}(x,t)\) and \(h_{j}(x,t)\) are \(s\) inequality and \(q\) equality constraint functions, respectively.  
\(\bullet\) Tchebycheff (TCH) approach  
Referring to the standard Tchebycheff formula in the static multi-objective problem, under the condition of adding time, a dynamic Tchebycheff scalar optimization problem can be described as:  
\[\left\{\begin{array}{l}\min\limits_{x\in\Omega}g^{\text{tch}}(F(x,t)\left| \lambda,z^{*}(t)\right.)=\max\limits_{1\leq i\leq m}\{\lambda_{i}\left|f_{i}( x,t)-z^{*}(t)\right|\},\\ \text{s. t. }g_{i}(x,t)\leq 0,i=1,2,...,s;h_{j}(x,t)=0,j=1,2,...,q,\end{array}\right. \tag{4}\]  
where \(\lambda=(\lambda_{1},\lambda_{2},...,\lambda_{m})^{T}\) is the same as the WS approach, \(t\) is a time variable, and \(z^{*}(t)=(z_{1}^{*}(t),...,z_{m}^{*}(t))^{T}\) is a reference point. That is, for each \(i=1,...,m\), \(z_{i}^{*}(t)=\min\{f_{i}(x,t)\left|g(x,t)\leq 0;h(x,t)=0;x\in\Omega\right.\}\). For each Pareto optimal solution \(x^{*}\) of MOPs in Formula (2), there is a corresponding weight vector \(\lambda\), so that \(x^{*}\) is the optimal solution of Formula (4). Each optimal solution of Formula (4) is also the Pareto optimal solution [23] of MOPs in Formula (2). Therefore, different Pareto optimal solutions can be obtained by changing the weight vectors. Thus, MOPs can be decomposed and transformed into multiple Tchebycheff scalar optimization subproblems with different weight vectors.  
\(\bullet\) Penalty-based boundary intersection (PBI) approach  
A dynamic PBI scalar optimization problem can be described as:  
\[\left\{\begin{array}{l}\min\limits_{x\in\Omega}g^{\text{pbi}}(x,t\left| \lambda,z^{*}(t)\right.)=d_{1}+\theta d_{2},\\ d_{1}=\frac{\left\|(z^{*}(t)-F(x,t))^{T}\lambda\right\|}{\left\|\lambda \right\|},d_{2}=\left\|F(x,t)-(z^{*}(t)-d_{1}\lambda)\right\|,\\ \text{s. t. }g_{i}(x,t)\leq 0,i=1,2,...,s;h_{j}(x,t)=0,j=1,2,...,q,\end{array}\right. \tag{5}\]  
where \(g^{\text{pbi}}(x,t\left|\lambda,z^{*}(t)\right.)\) is dynamic scalar objective function. \(z^{*}\) is the ideal reference point as defined in (4), and \(\theta\) is the default penalty parameter. \(F(x,t)\) is the evaluation function in Formula (2), \(d_{1}\) and \(d_{2}\) are two distance values [23], \(t\) is a time variable. According to previous experience [23], when using the same distributed weighted vectors, the PBI approach has a slight advantage over the TCH approach when solving MOPs with more than two objectives. However, the benefits come with the price that the penalty parameter \(\theta\) needs to be adjusted properly.