# Error-Function-Based Penalized Quantile Regression in the Linear Mixed Model
## 2 Model and Algorithm
In this paper, we utilize the IRW-pADMM algorithm, which combines the proximal ADMM algorithm with the iterative reweighted \(L_{1}\) algorithm. Consider the univariate quantile regression method with ERF regularization:  
\[Q\big{(}\tau\mid x_{ij},z_{ij},\alpha_{i}\big{)}=x_{ij}^{\mathrm{T}}\beta+z_{ ij}^{\mathrm{T}}\alpha_{i}, \tag{2}\]7  
where \(\beta=(\beta_{1},\ldots,\beta_{p})^{\mathrm{T}}\), \(x_{i}^{\mathrm{T}}\in\mathbb{R}^{p}\) is the \(i\)-th row of the design matrix \(X=(x_{1},x_{2},\ldots,x_{n})^{\mathrm{T}}\), and \(Y=(Y_{1},\ldots,Y_{n})^{\mathrm{T}}\in\mathbb{R}^{n}\) is the response vector. The IRW-pADMM algorithm is summarized in Algorithm 1. Here, \(\mathrm{shrink}(x,\alpha)=\mathrm{sgn}(x)\max(|x|-\alpha,0)\) is the soft thresholding operator and we define  
\[\mathrm{Prox}_{x_{p\tau}}(x,\alpha)=\left\{ \begin{array}{ll}x-\frac{\tau}{a},&x>\frac{\tau}{a},\\ 0,&\frac{\tau-1}{a}<x,\frac{\tau}{a},\\ x-\frac{\tau-1}{a},&x<\frac{\tau-1}{a},\end{array} \right.\]  
where the parameter \(\delta>0\) is the penalty parameter in the augmented Lagrangian of the ADMM algorithm, which affects the convergence speed of the model. \(\gamma>0\) is the step size parameter that regulates the update of \(\theta\). Based on the research conducted by [PERSON] et al., \(\eta\geq\lambda_{\max}\big{(}X^{\mathrm{T}}X\big{)}\) is used in the update of \(\beta\) to ensure that the matrix in the proximal term  
\[S=\sigma\Big{(}\eta I_{p}-X^{\mathrm{T}}X\Big{)}\]  
is positive semi-definite, and \(\lambda_{\max}\big{(}X^{\mathrm{T}}X\big{)}\) denotes the largest eigenvalue of the matrix \(X^{\mathrm{T}}X\)[20]. ```
1:Input:\(X\in\mathbb{R}^{n\times p}\), \(Y\in\mathbb{R}^{n}\), \(\sigma\), \(\lambda\), \(\gamma\), \(\tau\), \(\delta\), maxouter, maximner
2:Initialize:\(\hat{\beta}^{(0)}\), \(\hat{z}^{(0)}\), \(\hat{\theta}^{(0)}\), \(k\gets 0\)
3:while\(k<\) maxouter and \(\|\hat{\beta}^{(k+1)}-\hat{\beta}^{(k)}\|_{2}>\epsilon_{1}\)do
4:\(\omega^{(k)}\leftarrow\exp\left(-\bigg{(}\frac{\hat{\beta}^{(k)}}{\sigma} \bigg{)}^{2}\right)\)
5:\(l\gets 0\), \(\beta^{(l)}\leftarrow\hat{\beta}^{(k)}\)
6:while\(l<\) maxinner and \(\|\beta^{(l+1)}-\beta^{(l)}\|_{2}>\epsilon_{2}\)do
7:\(\beta^{(l+1)}\leftarrow\left(\mathrm{shrink}\Big{(}\beta_{j}^{(l)}+\frac{1}{ \delta q}X_{j}^{\mathrm{T}}\Big{(}\theta^{(l)}+\delta Y-\delta X\beta^{(l)}- \delta z^{(l)}\Big{)},\frac{\lambda\omega_{j}^{(l)}}{\delta q}\Big{)}\right)_{ 1\leq l\leq p}\)
8:\(z^{(l+1)}\leftarrow\left(\mathrm{Prox}_{\rho_{\tau}}\Big{(}Y_{i}-x_{i}^{\mathrm{ T}}\hat{\beta}^{(l+1)}+\frac{1}{2}\theta_{i}^{(l)},\delta\Big{)}\right)_{1\leq l \leq n}\)
9:\(\theta^{(l+1)}\leftarrow\theta^{(l)}-\gamma\delta\Big{(}X\beta^{(l+1)}+z^{(l+ 1)}-Y\Big{)}\)
10:\(l\gets l+1\)
11:endwhile
12:\(\hat{\beta}^{(k+1)}\leftarrow\beta^{(l)}\)
13:\(k\gets k+1\)
14:endwhile
15:return\(\hat{\beta}^{(k)}\)
```  
**Algorithm 1** IRW-pADMM algorithm. We present the two-step DERF-QR Algorithm in Algorithm 2 for the doubly penalized quantile regression method Equation (4). In Algorithm 2, we first initialize the fixed-effect coefficients as \(\hat{\beta}_{i}^{(0)}=0\), and then compute the initial estimates of the random-effect coefficients \(\hat{\alpha}^{(0)}\) using the IRW-pADMM algorithm. Subsequently, we alternately update the fixed- and random-effect coefficients by fixing one set and optimizing the other using the IRW-pADMM algorithm. Specifically, at each iteration \(k\), the fixed-effect coefficients are updated by solving \(\hat{\beta}^{(k+1)}=\arg\min_{\beta}L(\beta,\hat{\mathbb{A}}^{(k)})\) where the response variable is adjusted as \(Y_{ij}{}^{(k)}=Y_{ij}-z_{ij}\bar{\mathbb{A}}_{i}^{(k)}\). The updated \(\hat{\beta}^{(k+1)}\) is then substituted into the model to update the random-effects coefficients by solving \(\hat{\mathbb{A}}^{(k+1)}=\arg\min_{\mathbb{A}}L(\hat{\beta}^{(k+1)},\alpha)\). The residuals are recalculated as \(r_{ij}{}^{(k+1)}=Y_{ij}-x_{ij}\bar{\mathbb{A}}^{(k+1)}\).