## 4 ADMM Formulation and Solution
### Penalty Parameter Selection  
Choosing the penalty parameters properly is very important in ADMM. The values of penalty parameters are decreased or increased depending on the values of some predefined tolerances. Different methods can be used choose the penalty parameters, such as hit-and-trial, etc. Another method is to relate the values of the penalty parameters to iteration numbers so that the values of penalty parameters increase or decrease (from the initially \begin{table}
\begin{tabular}{c l l} \hline \hline
**Input**: & & \\ \multicolumn{3}{l}{Step (1) **Initialize:**\(\mathbf{r}_{l}^{0},\mathbf{s}_{r}^{0},\mathbf{u}^{0},\mathbf{v}^{0},\mathbf{w}^{0}, \rho_{1},\rho_{2},\rho_{3}\), and} \\ \(\epsilon_{1}^{prt},\epsilon_{2}^{pri},\epsilon_{2}^{pri},\epsilon_{2}^{dual},\,m=1\). & \\ \multicolumn{3}{l}{Step (2) **While** the termination criteria, Equation (34), are not satisfied, do} \\ \multicolumn{3}{l}{Step (3) Update \(\mathbf{r}_{l}^{m+1}\) using Equation (25)} \\ \multicolumn{3}{l}{Step (4) Update \(\mathbf{s}_{r}^{m+1}\) using Equation (29)} \\ \multicolumn{3}{l}{Step (5) Update \(\mathbf{u}^{m+1}\) using Equation (23c)} \\ \multicolumn{3}{l}{Step (6) Update \(\mathbf{v}^{m+1}\) using Equation (23d)} \\ \multicolumn{3}{l}{Step (7) Update \(\mathbf{w}^{m+1}\) using Equation (23e)} \\ \multicolumn{3}{l}{Step (8) \(m=m+1\)} \\ \multicolumn{3}{l}{Step (9) **End while**} \\
**Output**: & \\ \hline \hline \end{tabular}  
For clarity, a list of symbols, and their dimensions and descriptions, is provided in Table 2.  
\end{table} 
Table 2: List of symbols. \begin{table}
\begin{tabular}{c l l} \hline \hline
**Symbol** & **Dimension** & **Description** \\ \hline M & \(1\times 1\) & Number of antennas \\ N & \(1\times 1\) & Number of samples \\ K & \(1\times 1\) & Number of sidelobes \\ d & \(1\times 1\) & Antenna inter-element spacing \\ \(\mathbf{I}_{N}\) & \(N\times N\) & Identity matrix \\ \(\lambda\) & \(1\times 1\) & Wavelength \\ \(s_{m}(n)\) & \(1\times 1\) & \(n\)th sample of a discrete waveform \\ \(\mathbf{s}(n)\) & \(M\times 1\) & \(n\)th samples of the waveforms transmitted by all antennas \\ \(\mathbf{S}\) & \(M\times N\) & Space-time transmit waveform matrix \\ \(\mathbf{\bar{s}}\) & \(MN\times 1\) & Vector version of \(\mathbf{s}\) \\ \(\mathbf{\bar{s}}_{r}\) & \(2 MN\times 1\) & Real-valued version of \(\mathbf{\bar{s}}\) \\ \(\mathbf{x}_{R}\) & \(N\times 1\) & Desired radar waveform \\ \(\mathbf{x}_{C}\) & \(N\times 1\) & Desired communication waveform \\ \(\mathbf{X}\) & \(2\times N\) & Combination of desired communication waveform as a matrix \\ \(\mathbf{\bar{x}}\) & \(2N\times 1\) & Vector version of \(\mathbf{X}\) \\ \(\mathbf{\bar{x}}_{r}\) & \(4N\times 1\) & Real-valued version of \(\mathbf{\bar{x}}\) \\ \(\mathbf{a}(\theta_{R})\) & \(M\times 1\) & Steering vector in radar direction \\ \(\mathbf{a}(\theta_{C})\) & \(M\times 1\) & Steering vector in communication direction \\ \(\mathbf{A}(\Theta)\) & \(M\times 2\) & Combination of \(\mathbf{a}(\theta_{R}\) and \(\mathbf{a}(\theta_{C}\) \\ \(\mathbf{\overline{A}}(\Theta)\) & \(MN\times 2N\) & Vector version of \(\mathbf{A}(\Theta)\) \\ \(\mathbf{\overline{A}}_{r}(\Theta)\) & \(2 MN\times 4N\) & Real-valued version of \(\mathbf{\overline{A}}(\Theta)\) \\ \(\mathbf{A}\left(\widehat{\Theta}\right)\) & \(M\times K\) & Combination of sidelobe steering vectors \\ \(\mathbf{\overline{A}}\left(\widehat{\Theta}\right)\) & \(MN\times KN\) & Vector version of \(\mathbf{A}\left(\widehat{\Theta}\right)\) \\ \(\mathbf{\overline{A}}_{r}\left(\widehat{\Theta}\right)\) & \(2 MN\times 2 KN\) & Real-valued version of \(\mathbf{\overline{A}}\left(\widehat{\Theta}\right)\) \\ \(\mathbf{u}\) & \(4N\times 1\) & Dual variable \\ \(\mathbf{v}\) & \(MN\times 1\) & Dual variable \\ \(\mathbf{w}\) & \(2 MN\times 1\) & Dual variable \\ \(\eta,\mu\) & \(1\times 1\) & Positive constants \\ \(\rho_{1},\rho_{2},\rho_{3}\) & \(1\times 1\) & Penalty parameters \\ \hline \hline \end{tabular}
\end{table} 
Table 2: List of symbols. defined value) in steps.