## 1 Introduction
The CLASSO-RLS algorithm selects \(NL\) optimized secondary loudspeakers from \(NS\) candidate secondary loudspeakers. The CLASSO-RLS algorithm ensures a relatively large noise reduction in the target area as well as at the constraint points when selecting the sparse penalty parameter \(\gamma_{CLASSO}\)The maximum noise reduction in the target area corresponds to a particular sparse penalty parameter. A small value of noise reduction \(\Delta_{NR}\) will be neglected in this paper, so that several sparse penalty parameters corresponding to relatively large noise reduction can be obtained within a certain noise reduction range \([NR_{\text{max}}-\Delta_{NR},\ NR_{\text{max}}]\). Subsequently, the parameter with the largest noise reduction at the constraint points is selected as the final sparse penalty parameter \(\gamma_{\text{CLASSSO}}\). The detailed procedure for selecting \(\gamma_{CLASSSO}\) is as follows:  
Step 1: \(\gamma_{\text{n}}\) is the possible sparse penalty value and \(N_{\gamma}\) possible sparse penalty parameters are selected evenly in the selected value interval (0, \(\left\|\mathbf{Z}_{left}\mathbf{Z}_{right}\mathbf{q}_{p}\right\|_{\infty}\)) to form the possible sparse penalty value set \(\left\{\gamma_{\text{n}}\right\}_{\text{n}=1}^{N_{\gamma}}\) (dimension \(N_{\gamma}\times 1\)). Step 2: Using the CVX toolbox [27] to solve Equation (7) and obtain the solution \(\mathbf{q}_{s}\), the secondary loudspeakers with the largest \(NL\) amplitude in \(\mathbf{q}_{s}\) are selected as the chosen secondary loudspeakers for each \(\gamma_{\text{n}}\). Equation (6) is used to calculate the strength vector (dimension \(NL\times 1\)) of the chosen secondary loudspeakers \(\mathbf{q}_{s_{-}\gamma_{\text{n}}}\), and calculate the target area noise reduction \(NR_{\gamma_{\text{n}}}\):  
\[NR_{\gamma_{\text{n}}}=10\log\frac{\left\|\mathbf{Z}_{p_{c},sca}\mathbf{q}_{p} \right\|_{2}^{2}}{\left\|\mathbf{Z}_{p_{c},sca}\mathbf{q}_{p}+\mathbf{Z}_{sc _{-}\gamma_{\text{n}}}\mathbf{q}_{s_{-}\gamma_{\text{n}}}\right\|_{2}^{2}} \tag{8}\]  
where \(\mathbf{Z}_{sc_{-}\gamma_{\text{n}}}\) is the matrix (dimension \((N_{e1}+N_{e2})\times NL\)) of transfer functions from the chosen secondary loudspeakers to the target area for \(\gamma_{\text{n}}\). The calculation of the constraint point noise reduction \(NR_{c_{-}\gamma_{\text{n}}}\) is carried out using:  
\[NR_{c_{-}\gamma_{\text{n}}}=10\log\frac{\left\|\mathbf{Z}_{pc,sca}\mathbf{q}_{ p}\right\|_{2}^{2}}{\left\|\mathbf{Z}_{pc,sca}\mathbf{q}_{p}+\mathbf{Z}_{sc_{-} \gamma_{\text{n}}}\mathbf{q}_{s_{-}\gamma_{\text{n}}}\right\|_{2}^{2}} \tag{9}\]  
where \(\mathbf{Z}_{pc,sca}\) is the scattered sound matrix (dimension \(N_{c}\times 1\)) of the constraint points, \(\mathbf{Z}_{sc_{-}\gamma_{\text{n}}}\) is the transfer function matrix (dimension \(N_{c}\times NL\)) from the chosen secondary loudspeakers to the constraint points for \(\gamma_{\text{n}}\), and \(\mathbf{Z}_{sc}\) is the transfer function matrix (dimension \(N_{c}\times NS\)) from the candidate secondary loudspeakers to the constraint points; \(\mathbf{Z}_{sc_{-}\gamma_{\text{n}}}\) is selected from \(\mathbf{Z}_{sc}\). Here, the target area noise reduction set \(\mathbf{NR}_{\gamma_{\text{n}}}=\left[ \begin{array}{cc}NR_{\gamma_{1}}& \ldots&NR_{\gamma_{N_{\gamma}}}\end{array} \right]\) (dimension \(N_{\gamma}\times 1\)) and the constraint point noise reduction set \(\mathbf{NR}_{c_{-}\gamma_{\text{n}}}=\left[ \begin{array}{cc}NR_{c_{-}\gamma_ {1}}&\ldots&NR_{c_{-}\gamma_{N_{\gamma}}}\end{array} \right]\) (dimension \(N_{\gamma}\times 1\)) are formed. An important point of consideration after solving Equation (7) is the number of non-zero entries of \(\mathbf{q}_{s}\). If this is less than \(NL\), step 2 can be concluded. Step 3: Let \(NR_{\text{max}}=\text{max}(\mathbf{NR}_{\gamma_{\text{n}}})\); \(\Delta_{NR}\) is a small value of noise reduction artificially defined and in this paper \(\Delta_{NR}\)= 1 dB.