## 3 Proposed ATMALS-GSK Algorithm
### Atmals-Gsk
#### 3.2.6 Overall Operation of ATMALS-GSK  
The overall pseudo-code of the ATMALS-GSK algorithm is outlined in Algorithm 4. It integrates adaptive parameter tuning, memory-guided learning, and a probabilistic local search mechanism to enhance solution quality over generations. Key parameters (\(K\), \(K_{f}\), \(K_{r}\), \(P\), and \(P_{LS}\)) are dynamically selected using a Gaussian-enhanced probability scheme and powered RW S. The algorithm iteratively partitions the population into Junior and Senior groups, applies corresponding update rules, and performs local refinements to exploit promising areas of the solution space. ```
1:Inputs:
2:- MaxGen: Maximum generations
3:- PopSize: Population size
4:- ProbSize: Problem dimension
5:- Fitness(): Objective function to minimize
6:- \(K_{VALUES}\leftarrow[1,2,3,\ldots,30]\)
7:- \(K_{VALUES}\leftarrow[0.2,0.3,\ldots,0.8]\)
8:- \(K_{VALUES}\leftarrow[0.8,0.81,\ldots,1.0]\)
9:- \(P_{VALUES}\leftarrow[0.05,0.06,\ldots,0.2]\)
10:- \(PLS_{VALUES}\leftarrow[0.1,0.11,\ldots,0.2]\)
11: Initialize Probabilities for \(K\), \(K_{f}\), \(K_{r}\), \(P\), and \(P_{SL}\), using uniform distribution
12:Initialize
13:\(RP_{G}\gets 1\),
14:\(N_{LS}\gets 0.05\times PopSize\),
15:\(\alpha\gets 0.97\),
16:\(\delta\gets 0.05\times(\max(VALUES)-\min(VALUES))\)
17:Main ATMALS-GSK Algorithm:
18: Initialize population of PopSize individuals randomly
19:for1 generation \(G\) = 1 to MaxGen do:
20: Evaluate Fitness of all individuals
21:for2 each parameter in \(\{K\), \(K_{f}\), \(K_{r}\), \(P\), \(P_{LS}\}\) do:
22: Compute Gaussian enhancement rate based on Equations (3) and (4)
23: Update selection probabilities using Equation (5)
24: Apply powered RWS using Equations (6) and (7) to choose parameter values
25:endfor2
26:\(RP_{G}\leftarrow\) Adjust \(RP_{G}\) based on generation \(G\) (gradual increase)
27: Partition individuals into Junior and Senior based on adaptive \(Kr\) value
28:for3 each individual \(i\) in population do:
29:if\(i\) is in Junior group then:
30: Update D(junior) using junior learning rule (based on \(K\))
31:elseif\(i\) is in Senior group then:
32: Update D(senior) using senior learning rule (based on \(K\))
33:endfor3
34: Update \(Range_{LS}(G)\) using Equation (8)
35: Select \(N_{LS}\) individuals randomly
36:for4 each selected individual in \(N_{LS}\) do:
37:for5 each dimension \(d\) do:
38: Apply adaptive local search with probability of \(P_{LS}/ProbSize\)
39:endfor5
40:endfor4
41: Update historical memory of parameters' fitness improvements
42: Update the global best solution found so far
43:endfor1
44:Best solution found and its fitness value

## 4 Performance Evaluation

In this section, the detailed results of the ATMALS-GSK algorithm are provided and compared against the original GSK [16] and advanced GSK versions, including AGSK [46], APGSK [45], FDBAGSK [36], and eGSK [49], as well as recent metaheuristic algorithms, including IADE [50], GJO-JOS [51], QSCA [52], cSM [53], and FOWFO [54]. ### Settings

In the original GSK algorithm, the parameters \(K\), \(K_{f}\), \(K_{r}\), and \(P\) were tuned empirically using a trial-and-error process. This involved running multiple experiments on benchmark functions and selecting the parameter values that yielded the best average performance. However, this method is problem-specific and lacks generalizability. Therefore, the values tuned for one problem may not transfer well to different optimization problems, especially technical or real-world ones. In contrast, the ATMALS-GSK algorithm adaptively tunes these parameters during runtime based on historical performance, making it more robust and broadly applicable across different problem domains. The effectiveness of the proposed ATMALS-GSK algorithm is assessed using the 22 functions from the CEC 2011 and 30 functions from the CEC 2017 test suites (including unimodal functions, multimodal functions, hybrid functions, and composition functions). The full details of these benchmark problems can be found in the technical reports of CEC 2011 [55] and CEC 2017 [56].