## 1 Introduction
By iterating through the target area noise reduction set \(\mathbf{NR}_{\gamma_{\text{n}}}\), if \(NR_{\gamma_{\text{n}}}\geq NR_{\text{max}}-\Delta_{NR}\), the corresponding \(\gamma_{\text{n}}\) is selected to form the alternative parameter set of \(N_{a}\) values \(\mathbf{\gamma}_{a}=\left[\gamma_{1},\gamma_{2},\ldots,\gamma_{N_{a}}\right]\) (dimension \(N_{a}\times 1\)) and the corresponding constraint point noise reduction \(NR_{c_{-}\gamma_{\text{n}}}\) is selected to form the set of \(N_{a}\) values \(\mathbf{NR}_{c_{-}\gamma_{a}}=\left[ \begin{array}{cc}NR_{c_{-}\gamma_{1}}& \ldots&NR_{c_{-}\gamma_{\text{n}}}\end{array} \right]\) (dimension \(N_{a}\times 1\)). Step 4: Let \(NR_{c0}=\text{max}(\mathbf{NR}_{c_{-}\gamma_{\text{n}}})\); the \(\gamma_{a}\) corresponding to the maximum constraint point noise reduction \(NR_{c0}\) is selected as the final sparse penalty parameter \(\gamma_{CLASSSO}\). The CLASSSO-RLS algorithm is divided into two stages. In the first stage, the value of \(\gamma_{CLASSSO}\) is selected and the optimized \(NL\) secondary loudspeakers are obtained by solving Equation (7) using the CVX toolbox. In the second stage, the strength vector of the optimized secondary loudspeakers \(\mathbf{q}_{s_{-}\gamma_{\text{CLASSSO}}}\) is solved using Equation (6). The complete computational description of the CLASSSO-RLS algorithm is summarized in Algorithm 1. The main symbols are introduced in Table 1 in Appendix A. ```
Input: the matrix of scattered acoustic transfer functions of \(N_{c1}\) error points in target area \(1\)\(\mathbf{Z}_{p\ell\_sca}\). Input: the matrix of scattered acoustic transfer functions of \(N_{c2}\) error points in target area \(2\)\(\mathbf{Z}_{p\ell\_sca}\). Input: the matrix of scattered acoustic transfer functions of \(N_{c}\) constraint points \(\mathbf{Z}_{p\ell\_sca}\). Input: the matrix of transfer functions from \(NS\) candidate secondary loudspeakers to \(N_{c1}\) error points in target area \(1\)\(\mathbf{Z}_{s\ell 1}\). Input: the matrix of transfer functions from \(NS\) candidate secondary loudspeakers to \(N_{c2}\) error points in target area \(2\)\(\mathbf{Z}_{s\ell 2}\). Input: the matrix of transfer functions from \(NS\) candidate secondary loudspeakers to \(N_{c}\) conatraint points \(\mathbf{Z}_{s\ell}\). Input: the strength vector of one incident plane wave \(\mathbf{q}_{p}\). Output: the strength vector of optimized secondary loudspeakers \(\mathbf{q}_{\_s\_7 CLASSO}\). Step 1: Select \(N_{\gamma}\) possible parameters 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 \(\left\{\gamma_{n}\right\}_{n=1}^{N_{\gamma}}\). Step 2: Initialize the target area noise reduction set \(\mathbf{NR}_{\_}{\gamma_{n}}\) and the constraint point noise reduction set \(\mathbf{NR}_{\_}{\gamma_{n}}\), so that their elements are 0. Step 3: for \(\text{n}=1,2,\ldots,N_{\gamma}\). \(\gamma_{\_CLASSO}=\gamma_{n}\), Use the CVX toolbox to solve Equation (7), the secondary loudspeakers with the \(NL\) largest amplitude in \(\mathbf{q}_{\_}{\text{s}}\) are selected. if \(\left\|\mathbf{q}_{\_}{\text{s}}\right\|_{\_}{0}\geq NL\) Use Equation (6) to calculate the strength vector of the chosen secondary loudspeakers \(\mathbf{q}_{\_}{\text{s},\gamma_{\_}{\gamma_{\_}{\gamma_{\_}{\gamma_{\_}{ \gamma_{\_}{\gamma_{\_}{\gamma_{\_}{\gamma_{\_}\gamma_{\_{\_}}}}}}}}}}\), use Equations (8) and (9) to calculate the target area noise reduction \(\overline{NR}_{\_}{\gamma_{n}}\) and the constraint point noise reduction \(\mathbf{NR}_{\_}{\_}{\_}{\text{c},\gamma_{\_}{\gamma_{\_}{\gamma_{\_}{\gamma_{ \_}{\gamma_{\_}{\gamma_{\_}{\gamma_{\_}{\gamma_{\_{\_}}}}}}}}}}\).