CSA-CFGD¶

Implements CSA-CFGD, a cuckoo search optimizer whose moves are driven by the Caputo fractional gradient.

The method replaces the integer-order gradient inside the cuckoo search algorithm with a Caputo fractional gradient of order \(\alpha \in (0,1)\). The fractional gradient at iteration \(k\) is taken about a moving lower terminal \(c_k\) (the mean of three randomly perturbed solutions for the global walk, or the worst nest for the local walk) and evaluated by Gauss-Jacobi quadrature. New nests are then generated by Levy-flight global random walks and biased local random walks that descend along this fractional gradient of the best solution, combining the global exploration of cuckoo search with the curvature-aware descent of fractional calculus.

\[ \begin{aligned} {}^{C}\nabla^{\alpha}_x f(x^k)_i &= C_\alpha^k \sum_{m=0}^{M} \lambda_m\, f'_i\!\big[\Delta_k^i(1+u_m)+c_k^i\big] + C_\alpha^k\,\gamma_k\,(x_k^i-c_k^i)\sum_{m=0}^{M} \lambda_m\, f''_i\!\big[\Delta_k^i(1+u_m)+c_k^i\big] \\ x_j^{k+1} &= x_j^k + \beta_k \otimes \mathrm{Levy}(\lambda)\; {}^{C}\nabla^{\alpha}_x f(x_{\mathrm{best}}^k) \\ x_j^{k+1} &= x_j^k + \beta s \otimes H(p_a - v) \otimes {}^{C}\nabla^{\alpha}_x f(x_{\mathrm{best}}^k) \end{aligned} \]

where \(x_j\) is nest (solution) \(j\), \(x_{\mathrm{best}}\) the current best nest, \({}^{C}\nabla^{\alpha}_x f\) the Caputo fractional gradient of the objective \(f\), \(\alpha\in(0,1)\) the fractional order, \(c_k\) the per-coordinate lower terminal, \(\Delta_k^i = x_k^i - c_k^i\), \(C_\alpha^k = (1-\alpha)\,2^{-(1-\alpha)}\) the normalizing constant, \(\gamma_k\) a curvature weight, \(\{\lambda_m, u_m\}\) Gauss-Jacobi quadrature weights and nodes, \(\beta_k,\beta\) step scales, \(\mathrm{Levy}(\lambda)\) a Levy-flight step, \(H\) the Heaviside step, \(p_a\) the discovery probability, \(v\) a uniform random number, and \(\otimes\) elementwise multiplication. The first line is the quadrature form of the core CFGD update \(x^{k+1}=x^k-\eta_k\,{}^{C}\nabla^{\alpha}_x f(x^k)\).

Reference: Abduljlil Ali Abduljlil Habeb, Ningbo Zhu, Mundher Mohammed Taresh, Talal Ahmed Ali Ali, "Deep ocular tumor classification model using cuckoo search algorithm and Caputo fractional gradient descent", PeerJ Computer Science 2024. https://doi.org/10.7717/peerj-cs.1923

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