Table 1 AEA-SD program flow.

Input: x, f_s, Iteration limit K, optimal threshold σ, current iteration k

1: k = 1; n = 0; ξ_n = x; Fit⁰_best = 1E6;

2: while (k < K&&||ξ_n||₂² > σ)

3: Determine the basic parameters of g-atom by reference parameters initialization algorithm

pare⁰ = [f ϕ d₁ d₂ t₀ τ₁ τ₂ ε];

4: Initialize BA and produce the initial position of the bat swarm within the range [0.8 1.2]*pare0; \(P^{0} { = }\left\{ {P_{i}^{0} \left| {i = 1,2,...,q} \right.} \right\} = para^{0} + \lambda_{i}^{0} \times \left( {1.2*para^{0} - 0.8*para^{0} } \right)\)

5: Best matching atom gⁿ_best trained by BA and calculate the optimal fitness value \(Fit_{best}^{n}\) by \(\left| {\left\langle {x,g_{{i_{best} }} } \right\rangle } \right| = \max_{{i \in \left( {1,....n} \right)}} \left| {\left\langle {x,g_{i} } \right\rangle } \right|\)

6: If (\(Fit_{best}^{n} < Fit_{best}^{n - 1}\) )

7: update \(\xi_{{n{ + 1}}} = \xi_{n} - \left\langle {c_{best}^{n} * g_{best}^{n} \xi_{n} } \right\rangle * c_{best}^{n} * g_{best}^{n}\);

8: n = n + 1;

9: else

10: update \(Fit_{best}^{n} = Fit_{best}^{n - 1}\)

11: end

12: k = k + 1;

13: end

Output: ŝ = x-ξ_n