Table 3 Comparative performance and complexity summary of the proposed hybrid ADMM–Benders–continuation algorithm.
Algorithm | Average iterations | Avg. time per case (s) | Residual tolerance (\(\Vert \textbf{r}\Vert _2\)) | Stopping success rate (%) | Theoretical complexity |
|---|---|---|---|---|---|
Standard ADMM | 326 | 0.84 | \(<10^{-3}\) | 96.2 | \(\mathcal {O}(N T)\) |
Proposed Hybrid Scheme | 278 | 0.79 | \(<10^{-4}\) | 99.1 | \(\mathcal {O}(N T + K \log T)\) |
Continuation-only Variant | 301 | 0.82 | \(<10^{-3}\) | 97.4 | \(\mathcal {O}(N T)\) |
Hybrid + Warm-start | 255 | 0.76 | \(<10^{-4}\) | 99.3 | \(\mathcal {O}(N T + K)\) |
