# A Dynamic Parameter Tuning Strategy for Decomposition-Based Multi-Objective Evolutionary Algorithms
## 4 Discussion
### Performance Indicators  
This paper employs the inverted generational distance evaluation metric (IGD) and the hypervolume metric (HV) to evaluate multi-objective optimization algorithms.  
The IGD value indicates the minimum distance from the population to the ideal Pareto front obtained by the algorithm, i.e., the smaller the value of IGD, the closer the population is to the ideal Pareto front and is distributed in each part of the Pareto front, the better the algorithm performs. Therefore, the IGD value can be a good assessment of the convergence and diversity of the algorithm.  
Inverted generational distance (IGD):  
\[IGD(P^{*},P)=\frac{\sum_{\alpha\in P^{*}}d(v,P)}{|P^{*}|} \tag{8}\]  
The function \(d(v,P)\) represents the minimum Euclidean distance from the point v to P. The point P denotes the solution set of the algorithm to find the population. P\({}^{*}\) is a set of uniformly distributed points obtained from the ideal Pareto front, and \(|P^{*}|\) is the size of P\({}^{*}\).  
The HV value is used to comprehensively evaluate the convergence and diversity of the solutions obtained by the multi-objective optimization algorithm. The larger the HV value, the closer the current population is considered to be to the ideal Pareto frontier. However, the choice of reference point needs to be more careful, because it will greatly affect the accuracy of the HV value.  
\begin{table}
\begin{tabular}{c c c} \hline
**Problem** & **Objective Functions** & **Variable Bounds** \\ \hline \multirow{6}{*}{DTLZ3} & \(Min\ F=\{f_{1},f_{2},f_{3}\}\) & \\  & \(f_{1}=(1+g)cos(x_{1}\pi/2)cos(x_{2}\pi/2)\) & \\  & \(f_{2}=(1+g)cos(x_{1}\pi/2)sin(x_{1}\pi/2)\) & \(x_{i}e[0,1]\) \\  & \(f_{3}=(1+g)sin(x_{1}\pi/2)\) & \\  & \(g=100[|x_{m}|+\sum\limits_{x_{i}\in x_{M}}(x_{i}-0.5)^{2}-cos(20\pi(x_{i}-0.5) )]\) & \\ \hline \multirow{6}{*}{DTLZ4} & \(Min\ F=\{f_{1},f_{2},f_{3}\}\) & \\  & \(f_{1}=(1+g)cos(x_{1}^{\pi}\pi/2)cos(x_{2}^{\pi}\pi/2)\) & \\  & \(f_{2}=(1+g)cos(x_{1}^{\pi}\pi/2)sin(x_{2}^{\pi}\pi/2)\) & \\  & \(f_{3}=(1+g)sin(x_{1}^{\pi}\pi/2)\) & \\  & \(g=\sum\limits_{x_{i}\in x_{M}}(x_{i}-0.5)^{2}\) & \\ \hline \multirow{6}{*}{DTLZ5} & \(Min\ F=\{f_{1},f_{2},f_{3}\}\) & \\  & \(f_{1}=(1+g)cos(\theta_{1}\pi/2)cos(\theta_{2}\pi/2)\) & \\  & \(f_{2}=(1+g)cos(\theta_{1}\pi/2)sin(\theta_{2}\pi/2)\) & \\  & \(f_{3}=(1+g)sin(\theta_{1}\pi/2)\) & \\  & \(\theta_{i}=\frac{\pi}{4(1+g)}(1+2 gx_{i})\) & \\  & \(g=\sum\limits_{x_{i}\in x_{M}}x_{i}^{0.1}\) & \\ \hline \multirow{6}{*}{DTLZ6} & \(Min\ F=\{f_{1},f_{2},f_{3}\}\) & \\  & \(f_{1}=(1+g)cos(\theta_{1}\pi/2)cos(\theta_{2}\pi/2)\) & \\  & \(f_{2}=(1+g)cos(\theta_{1}\pi/2)sin(\theta_{2}\pi/2)\) & \\  & \(f_{3}=(1+g)sin(\theta_{1}\pi/2)\) & \\  & \(\theta_{i}=\frac{\pi}{4(1+g)}(1+2 gx_{i})\) & \\  & \(g=\sum\limits_{x_{i}\in x_{M}}x_{i}^{0.1}\) & \\ \hline \multirow{6}{*}{DTLZ7} & \(Min\ F=\{f_{1},f_{2},f_{3}\}\) & \\  & \(f_{1}=(1+g)cos(\theta_{1}\pi/2)cos(\theta_{2}\pi/2)\) & \\  & \(f_{2}=(1+g)cos(\theta_{1}\pi/2)sin(\theta_{2}\pi/2)\) & \\  & \(f_{3}=(1+g)sin(\theta_{1}\pi/2)\) & \\ \cline{1-1}  & \(\theta_{i}=\frac{\pi}{4(1+g)}(1+2 gx_{i})\) & \\ \cline{1-1}  & \(g=\sum\limits_{x_{i}\in x_{M}}x_{i}^{0.1}\) & \\ \hline \multirow{6}{*}{DTLZ6} & \(Min\ F=\{f_{1},f_{2},f_{3}\}\) & \\  & \(f_{1}=x_{1}\) & \\ \cline{1-1}  & \(f_{2}=x_{2}\) & \\ \cline{1-1}  & \(f_{3}=x_{3}\) & \\ \hline \end{tabular}
\end{table}
Table 2: Cont.  
Hypervolume (HV):  
\[HV(X,P)=\bigcup_{x\in X}^{X}v(x,P) \tag{9}\]  
In the set of non-occupied solutions \(X\), \(v(x,P)\) represents the hypervolume of the space formed by the solution \(x\) and the reference point \(P\). This is equivalent to the volume of the hypercube constructed with the line between the solution \(x\) and the reference point \(P\) as the diagonal.