# A Dynamic Parameter Tuning Strategy for Decomposition-Based Multi-Objective Evolutionary Algorithms
## 3 Suggested Strategies for Improvement
### Boundary-Crossing Strategy for Adaptive Tuning of Penalty Parameters  
In this paper, adaptive parameter tuning is used to regulate the convergence and diversity of the population, thus accelerating population convergence. From the above analysis, it can be seen that the size of the penalty parameter \(\theta\) is inextricably related to the current iteration period. At the beginning of the iteration, the penalty parameter \(\theta\) is low, the population undergoes a large number of substitution operations, and the population convergence increases, rapidly approaching the Pareto frontier. In the later stages of the iteration, the penalty parameter value \(\theta\) rises and the population dispersion and diversity increase so that the population does not miss any part of the Pareto front. Therefore, this paper makes the following improvements based on Equation (2):  
During the evolution of a population, if the penalty parameter \(\theta\) is too small, the solutions at the frontier are easily replaced by solutions closer to the center, which results in a decrease in population diversity. If the penalty parameter \(\theta\) is too large, the occurrence of the replacement operation in the population is not frequent, which significantly impacts the convergence speed of the population and hinders its ability to approach the Pareto frontier effectively. As a result, the penalty parameter's value is related to the current iteration period, as well as the uniformity of the current population distribution. Therefore, on the basis of the above tuning formula, the distribution density of the current population should be taken into account. Therefore, the uniform state of the subproblem is defined first:  
\[avg^{i}=\frac{nc^{i}-nc_{min}}{nc_{max}-nc_{min}} \tag{3}\]  
Among them, \(\text{avg}^{i}\in[0,1]\) is the uniform state of the distribution of solutions around the weight vector, the number of solutions to the weight vector is represented by \(\text{nc}^{i}\). The minimum number of solutions is represented by \(\text{nc}_{\text{min}}\), and \(\text{nc}_{\text{max}}\) represents the maximum number of solutions. For the solution \(x^{i}\), if the distance between the vector and weight vector \(i\) is less than the distance between its corresponding solution \(x^{i}\) and weight vector \(i\), it is considered to be distributed around weight vector \(i\), i.e.,  
\[nc^{i}=\left\{x^{i}\Big{|}d\Big{(}x^{i},w^{i}\Big{)}<d\Big{(}x^{i},w^{i}\Big{)}\right\} \tag{4}\]  
where \(\text{d}(x,w)\) denotes the perpendicular distance from the solution \(x\) to the weight vector \(w\), i.e.,  
\[d(x,w)=\|\overline{F}(x)-\frac{w^{\text{T}}\overline{F}(x)}{w^{\text{T}}w}\| \tag{5}\]  
where \(\overline{\text{F}}(x)\) is the normalized processing function for solving \(x\), which is treated as in Equation (6). \[ASF\big{(}x,w_{j}\big{)}=max_{i=1}^{m}\{\frac{1}{w_{j}|}\bigg{|}\frac{f_{i}(x)- z_{i}^{*}}{z_{i}^{\text{nad}}-z_{i}^{*}}\bigg{|}\} \tag{6}\]  
Since the final population sought has to balance convergence and diversity, if there are too many solutions distributed around the weight vector, it is too convergent, and the current population is not evenly distributed. During population evolution, if the penalty parameter \(\theta\) is too small, solutions at the boundary are easily replaced by solutions closer to the center. If the penalty parameter \(\theta\) is too large, the population is less susceptible to replacement operations. Therefore, this paper proposes to further adjust the penalty parameter \(\theta\) according to the current solution distribution state on the basis of tuning the parameter according to the current iteration state. When there are too many solutions distributed around the current weight vector, the penalty parameter is decreased to increase the convergence of the solutions; when there are too few solutions distributed around the weight vector, the penalty parameter is increased to increase the population diversity. \[\theta=\left\{