## 3 Details of the Proposed Algorithm
### Update of Primal Variable: \(\mathbf{x}\)
```  
**Algorithm 1** Update of Primary Variable \(\mathbf{x}\): Monotone Strategy [28]  
In this work, we integrate the APG method to enhance convergence speed [34], characterized by an additional extrapolation step to inject momentum from previous iterations  
\[\mathbf{u}^{(k+1)}=\mathbf{x}^{(k)}+\frac{\mu_{x}^{(k-1)}}{\mu_{x}^{(k)}}( \mathbf{w}_{x}^{(k)}-\mathbf{x}^{(k)})+\frac{\mu_{x}^{(k-1)}-1}{\mu_{x}^{(k)}} (\mathbf{x}^{(k)}-\mathbf{x}^{(k-1)}), \tag{25}\]  
where \(\mathbf{w}_{x}^{(k)}\) is the output of the proximal operator at \(\mathbf{u}^{(k)}\) from the previous iteration, and the extrapolation parameter \(\mu_{x}^{(k)}\) is updated by  
\[\mu_{x}^{(k+1)}=\frac{1+\sqrt{1+4\Big{(}\mu_{x}^{(k)}\Big{)}^{2}}}{2}. \tag{26}\]  
Accordingly, the proximal gradient step in (23) is adjusted to operate at the new extrapolation point \(\mathbf{u}^{(k)}\) as  
\[\mathbf{x}^{(k+1)}=\text{prox}_{f_{2},t_{x}}\Big{(}\mathbf{u}^{(k+1)}-t_{x} \
abla f_{1}(\mathbf{u}^{(k+1)})\Big{)}. \tag{27}\]  
In ill-conditioned problems or when the optimization landscapes comprise flat plateaus and narrow ravines, enforcing a monotonic decrease in the objective function can slow down the convergence speed [41]. To address this problem, a non-monotone update strategy that allows temporary increases in the objective function value across iterations is introduced. Specifically, we define a relaxation criterion using a convex combination of previous function values \(\{J(\mathbf{x}^{(0)})\), \(J(\mathbf{x}^{(1)},\cdots\ J(\mathbf{x}^{(k)})\}\):  
\[\bar{J}^{(k)}=\frac{\sum_{j=0}^{k}\varepsilon^{k-j}J(\mathbf{x}^{(j)})}{\sum_ {j=0}^{k}\varepsilon^{k-j}}, \tag{28}\]  
where \(\varepsilon\in[0,1)\) is an exponentially decreasing weight controlling the non-monotonicity level. If \(\varepsilon=0\), the strategy reverts to monotone. If \(\varepsilon=1\), \(\bar{J}^{(k)}\) is the average function value from all previous iterations. A similar criterion is also used in non-convex programming to facilitate faster convergence [42]. \(\bar{J}\) can also be computed efficiently in a recursive fashion:  
\[c^{(k+1)}=\varepsilon c^{(k)}+1,\quad\bar{J}^{(k+1)}=\frac{\varepsilon c^{(k)} \bar{J}^{(k)}+\bar{J}(\mathbf{x}^{(k+1)})}{c^{(k+1)}} \tag{29}\]  
with \(c^{(0)}=1\) and \(\bar{J}^{(0)}=\bar{J}(\mathbf{x}^{(0)})\). Since we do not require the objective function value to decrease monotonically as in Algorithm 1, the termination condition of the line search process can also be relaxed to reduce the number of inner iteration steps and allow a larger step size to be used. As illuminated in Lemma 2.1 of [23], the linear-quadratic approximation  
\[\
abla f_{1}(\mathbf{x}^{(k)})=\rho\boldsymbol{\Phi}^{H}(\mathbf{z}^{(k)}+ \boldsymbol{\Phi}\mathbf{x}^{(k)}-\mathbf{y}+\gamma^{(k)}/\rho). \tag{22}\]0  
can be used to indicate that a proper step-size \(t_{x}^{(k+1)}\) is found to ensure adherence to the Lipschitz continuity condition. The proposed non-monotone update strategy of primary variable \(\mathbf{x}\) is summarized in the Algorithm 2. To avoid the case that \(\mathbf{u}^{(k)}\) might be a bad extrapolation in the APG method, note that we also incorporate an additional normal proximal gradient step to serve as a monitor. This step, as specified in lines 7-9 of Algorithm 2, is activated only when the extrapolation point \(\mathbf{u}^{(k)}\) does not contribute to a decrease in the value of the relaxed objective function \(\bar{J}^{(k)}\). It ensures the algorithm maintains robust performance and convergence behavior without substantially increasing the computational burden. ```
0: Set \(\mathbf{x}^{(0)}=\mathbf{0}\), \(\mu_{x}^{(0)}=\mu_{x}^{(-1)}=1\), \(t_{0}>0\), and \(\eta_{x}>1\)....
0: Get \(\mathbf{x}^{(k-1)}\), \(\mathbf{x}^{(k-2)}\), \(\mathbf{z}^{(k-1)}\), \(\gamma^{(k-1)}\) and \(\bar{J}^{(k-1)}\) from previous iteration. 1: Compute the extrapolation point \(\mathbf{u}^{(k)}\) by (25) and the gradient \(\
abla f_{1}(\mathbf{u}^{(k)})\) by (22), respectively.