## 4 Numerical Simulations
### Simulation I: CIR Estimation in Single-Carrier System  
We simulate the UWA channel using the model from [43]. The channel has a total possible delay spread of 128 ms with a sampling interval \(\Delta\tau=0.25\,\mathrm{ms}\), leading to a CIR of \(N=512\) potential taps. \(N_{pa}=64\) of these taps are non-zero, highlighting the sparsity of the channel. The interarrival times for these non-zero taps are exponentially distributed, and their amplitudes follow a Rayleigh distribution with a variance that decreases as the delay increases. We employ the widely used Pseudo-Random Binary Sequence (PRBS) as the probe signal after binary phase-shift keying (BPSK) modulation. The probe symbol sequence length is set at \(M=\frac{1}{2}N=256\), rendering the problem underdetermined. In the impulsive noise environment, the Interference-to-Noise Ratio (INR) defined as \(\mathrm{INR}=10\log_{10}\left(\sigma_{1}^{2}/\sigma_{W}^{2}\right)\) is set to 30 dB and the occurrence probability \(q=5\times 10^{-2}\). To quantify estimation accuracy, we use the Normalized Mean-Square Deviation (NMSD) metric: \(\mathrm{NMSD}=20\log(||\mathbf{x}^{\ast}-\hat{\mathbf{x}}||_{2}/||\mathbf{x}^{ \ast}||_{2})\), where \(\mathbf{x}^{\ast}\) is the true CIR and \(\hat{\mathbf{x}}\) denotes its estimate. Regularization factors for algorithms FISTA, ADMM, and IPM-PCG are all set to \(\lambda=0.01\ \lambda_{\infty}\), where \(\lambda_{\infty}\Leftarrow||\mathbf{2\Phi}^{H}\mathbf{y}||_{\infty}\). For the mADMM and nmADMM algorithms, the regularization factor is set as \(\tau=1/(0.05\lambda_{\infty})\). Algorithms with a backtracking line search process, specifically FISTA, IPM-PCG, mADMM, and the proposed method, have their scale parameter \(\eta\) for stepsize reduction set to 1.5. The initial penalty parameter \(\rho\) for all ADMM-based algorithms is set to 1, and the absolute and relative tolerance thresholds for the stopping criterion are \(\epsilon_{\mathrm{abs}}=10^{-3}\) and \(\epsilon_{\mathrm{rel}}=10^{-2}\), respectively. The additional exponentially decreasing weight parameter which controls the non-monotonicity of the proposed nmADMM algorithm is set as \(\varepsilon=0.95\). The stopping criterion for the FISTA algorithm evaluates the relative change ratio of residuals between two consecutive iterations, given as \(||\mathbf{r}^{(k+1)}-\mathbf{r}^{(k)}||_{2}/||\mathbf{r}^{(k)}||_{2}\), where \(\mathbf{r}^{(k)}=\mathbf{y}-\mathbf{\Phi}\mathbf{x}^{(k)}\), and the tolerance for this ratio is \(\epsilon_{\mathrm{fista}}=10^{-3}\). The relative tolerance of the duality gap for the IPM-PCG algorithm is \(\epsilon_{\mathrm{ipm}}=10^{-3}\). Prior to deploying IPM-PCG, the problem is transitioned to the real domain using the method outlined in ref. [21]. All results are averaged over 100 independent runs. Figures 1 and 2 present an estimated CIR example in two noise environments at a Signal-to-Noise Ratio (SNR) of 10 dB, aligning with the practical conditions in UWA applications. The NMSD versus iteration curves are depicted in Figure 3. One can see that all algorithms achieve a similar NMSD of approximately -12 dB in the AWGN environment. The OMP algorithm can attain the lowest NMSD with precise prior information of sparsity and optimal stopping conditions. The second-order IPM-PCG algorithm offers the fastest convergence; however, its accuracy is compromised due to the problem conversion to the real domain and it has a much higher computational load than other algorithms. In the presence of impulsive noise, the performance of OMP, FISTA, ADMM, and IPM-PCG deteriorates drastically where the estimated CIR contains a large number of noisy taps that are supposed to be inactive. The accuracy of mADMM and nmADMM also decreases by less than 1 dB compared to their performance in the AWGN environment. Yet, they still effectively estimate the channel components. One can also observe that the proposed nmADMM algorithm exhibits a faster convergence speed than the mADMM algorithm, achieving convergence within approximately 20 iterations.