# Optimization of Slotted Steel Plate Shear Walls Based on Adaptive Genetic Algorithm
## 2 Design and Implementation of Adaptive Genetic Algorithm  
### Design of Adaptive Genetic Algorithm  
The canonical genetic algorithm (GA) framework comprises six fundamental operations: population initialization, fitness evaluation, selection, crossover, mutation, and generational replacement. The computational workflow initiates with the random generation of a population, where each individual solution is encoded as a chromosomal sequence. Subsequently, fitness values are calculated to quantify the solution quality. A selection operator identifies the high-fitness individuals as parental candidates for genetic recombination. The crossover operation probabilistically combines the parental chromosomes through simulated mating, generating novel offspring solutions. To enhance the population diversity, a mutation operator introduces stochastic gene-level modifications. The generational replacement phase then substitutes the parent population with the offspring cohort, completing one evolutionary cycle. This iterative process continues until the termination criteria are satisfied (e.g., maximum iterations or convergence of thresholds), progressively refining the solution quality. By emulating natural selection mechanisms, GAs effectively explore complex solution spaces. While standard GAs demonstrate multidisciplinary applicability, their inherent limitations include premature convergence to the local optima and computational inefficiency in the late-stage iterations. Contemporary research advancements have addressed these shortcomings through algorithmic enhancements, enabling broader theoretical and practical implementations across engineering domains [29; 30].  
This study proposes an improved adaptive genetic algorithm (IAGA) incorporating three key innovations: (1) dynamic probability adaptation, (2) an elitism strategy, and (3) convergence acceleration mechanisms, building upon the conventional genetic algorithm foundations. The validation through four benchmark function optimizations under identical computational conditions demonstrates that the IAGA effectively mitigates premature convergence, enhances the global search capability, and achieves accelerated convergence rates compared to standard genetic algorithms.  
Compared to conventional adaptive probability strategies [31], the proposed enhancement introduces critical modifications to Equation (1), specifically incorporating a boundary condition when \(f_{\text{max}}=f_{\text{avg}}\). This adaptation effectively prevents population homogeneity when implementing elitism strategies, thereby maintaining genetic diversity throughout the evolutionary iterations.  
\[\begin{split}\text{P}_{\text{c}}&=\begin{cases} \text{k}_{1},f_{\text{max}}=f_{\text{avg}}\\ \frac{\text{k}_{2}(f_{\text{max}}-f^{\prime})}{f_{\text{max}}-f_{\text{avg}}} +\text{k}_{3},f\geq f_{\text{avg}}\\ \text{k}_{4},f=f_{\text{avg}}\end{cases}\\ \text{P}_{\text{m}}&=\begin{cases}\text{k}_{5},f_{\text{max}}=f_{\text{avg}} \\ \frac{\text{k}_{6}(f_{\text{max}}-f)}{f_{\text{max}}-f_{\text{avg}}}+\text{k}_{7 },f\geq f_{\text{avg}}\\ \text{k}_{8},f=f_{\text{avg}}\end{cases}\end{split} \tag{1}\]  
In the equation, \(t^{\prime}\) represents the larger fitness value of the individuals to be crossed, \(f_{\text{max}}\) represents the maximum fitness value in the population, \(f_{\text{avg}}\) represents the average fitness value of the population, and \(f\) represents the fitness value of the mutated individual. The parameters \(\text{k}_{1}\), \(\text{k}_{2}\), \(\text{k}_{3}\), \(\text{k}_{4}\), \(\text{k}_{5}\), \(\text{k}_{6}\), \(\text{k}_{7}\), and \(\text{k}_{8}\) are all adaptive control parameters. The improved algorithm is shown in Figure 1.  
Figure 1: Adaptive genetic algorithm flow chart.