## 3 Proposed ATMALS-GSK Algorithm
### Atmals-Gsk
#### 3.2.1 Adaptive \(K\)  
The knowledge rate \(K\) serves as a key parameter in the GSK algorithm, controlling how much experience an individual accumulates throughout generations in the junior and senior phases. It directly affects the percentage of dimensions updated by each phase  
Figure 3: Overall flowchart of the ATMALS-GSK algorithm. over time. As shown in Figure 4 and detailed in Table 2, different values of \(K\) result in distinct learning dynamics. When \(K=1\), the transition between junior and senior phases is smooth and linear. The number of junior dimensions decreases steadily, while the number of senior dimensions increases at the same rate, perfectly balancing exploration early on and gradually shifting to exploitation. However, this linear behavior changes when \(K\
eq 1\): For \(K<1\) (e.g., \(K=0.5\)), the decline in junior updates is slow and non-linear, meaning the algorithm relies more heavily on exploration for a longer time. For instance, at 50% of the total generations, 71% of the dimensions are still updated by the junior scheme, with only 29% by the senior scheme. This results in gradual knowledge acquisition, allowing broader search of the solution space. Furthermore, for \(K>1\) (e.g., \(K=2\) or 5), the shift from junior to senior updates is faster and more aggressive. At \(K=2\), by halfway through the run, only 25% of dimensions are updated by the junior phase, and 75% by the senior. With \(K=5\), this shift is even more dramatic--only 3% of dimensions are updated via the junior scheme at the halfway point. This leads to faster convergence but may risk premature exploitation. Figure 4: Effect of \(K\) on the population of junior and senior individuals versus generation. These patterns reveal that \(K\) functions as a learning rate controller within the GSK algorithm. Lower values of \(K\) promote slower, more cautious learning by extending the influence of the junior phase and encouraging prolonged exploration of the solution space. In contrast, higher \(K\) values accelerate the learning process by shifting focus to the senior phase earlier, thus enabling faster and more targeted exploitation. The selected \(K\) value directly influences how updated dimensions are allocated between the junior and senior schemes in each generation. These distributions are initialized at the start and adapt automatically as the algorithm evolves, underscoring the critical role of \(K\) in striking the right balance between exploration and exploitation based on the specific characteristics of the problem at hand. In the proposed ATMALS-GSK algorithm, the range of \(K\) is considered within [5; 30] with a discretized level of 1. A historical weighted moving average function is used to adaptively tune the value of \(K\) based on an exponential function of the improvement rates of the different values of \(K\) in previous iterations. Weighted moving average [48] is a trend-following index used for smoothing the short-term variation of the parameter, which can be obtained on the basis of the past effectiveness of the different values of the parameter. The exponential weighted moving average function considers the weights that are assigned exponentially reducing for previous values, in which the closest values are of more importance.