# A Dynamic Parameter Tuning Strategy for Decomposition-Based Multi-Objective Evolutionary Algorithms
## 2 Related Work
### Effect of Penalizing Factor \(\theta\) on PBI  
[PERSON] and [PERSON] analyzed the effect of punishment factor \(\theta\) on PBI [6]. Specifically, when a solution does not meet the conditions, the objective function value of the solution is penalized by the PBI algorithm through the addition of a penalty term, and the size of this penalty term has a penalty factor to determine. If the penalty factor is excessive, the algorithm will penalize solutions that fail to meet the constraints, which may cause the algorithm to favor solutions that satisfy the constraints and therefore increase the diversity of the algorithm. On the contrary, if the penalty factor is smaller, the algorithm penalizes less for solutions that do not meet the constraints, which may cause the algorithm to be more inclined to find better solutions, i.e., solutions that may not meet the constraints, and therefore improve the convergence of the algorithm.  
Figures 1 and 2 show the effect of two different values of the penalty parameter \(\theta\) on convergence and diversity [6].  
The penalty factor \(\theta\) is set to \(1.0\) in Figure 1; \(\omega_{1}=(0.1,0.9)^{\text{T}}\) and \(\omega_{2}=(0.2,0.8)^{\text{T}}\) denote two neighboring boundary weight vectors. From Figure 1, B and C are the optimal points of \(\omega_{1},\omega_{2}\), two boundary weight vectors, respectively. Since \(\text{g}^{\text{pbi}}(\text{B}|\omega_{1},\text{z}^{*})=0.60\) is less than \(\text{g}^{\text{pbi}}(\text{B}|\omega_{1},\text{z}^{*})=0.75\), the solution of the subproblem for \(\omega_{1}\) with the associated sub-problem is replaced by C from B. So, when the penalty factor is small, the algorithm may find more non-dominated solutions because it is easier for the algorithm to find solutions that do not meet the constraints but have smaller objective function values. These non-dominated solutions may cause congestion in the Pareto front and thus reduce the diversity of the algorithms. In Figure 2, since the penalty factor \(\theta\) is set to a large 5.0, H and J are the optimal points of the two boundary weight vectors of \(\omega_{1},\omega_{2}\), respectively, but since the g\({}^{\text{pbi}}\) of J is larger than that of H, H will not be replaced by J. So, when the penalty factor is large, the algorithm may find fewer non-dominated solutions because it focuses more on those solutions that satisfy the constraints. These solutions satisfying the constraints may be more decentralized, thus increasing the diversity of the algorithms.  
And later in the paper [6], the adaptive penalization scheme (MOEA/D-APS) was proposed. MOEA/D-APS proposes adaptively adjusting the penalty factor \(\theta\) based on different search stages. In the early stage, the main optimization goal is to converge towards the real PF quickly, so the penalty factor \(\theta\) is set to a small value; In the subsequent stages, the penalty parameter is increasing, and at this point, the algorithm focuses more on the diversity of the population. APS uses this formula to achieve the above goals:  
\[\theta=\theta_{min}+(\theta_{max}-\theta_{min})\frac{it}{maxIteration} \tag{2}\]  
The variable it represents the current number of iterations, while maxIteration represents the total number of population iterations. \(\theta_{\text{min}}\) and \(\theta_{max}\) represent the upper and lower bounds of \(\theta\), respectively, and \(\theta\) takes the value 5.0; \(\theta_{\text{min}}=1.0,\theta_{\text{max}}=10.0\).