## 1 Introduction  
The active noise control (ANC) technique is based on the principle of superposition to cancel the primary noise by generating anti-noise with the same amplitude, but opposite phase as the primary noise [1; 2; 3]. Active noise control can also be applied to reduce the scattered sound from a reflecting object on which an incident wave impinges, thereby rendering the object invisible to the incident acoustic waves [4; 5]. Earlier studies utilized arrays of uniformly distributed secondary loudspeakers [4; 5; 6] to control noise. However, these strategies are limited to relatively low frequencies because of the following reason: as the wavelength of sound becomes smaller or the target area expands, the number of the secondary loudspeakers required to achieve good noise reduction in the target area increases, and the uniformly placed secondary loudspeaker array produces large sound amplification in other areas. In practical applications, the number of secondary loudspeakers and the available power are limited. How to achieve a good scattered sound control performance under the constrained condition is a critical issue that requires resolution [8]. The control performance in the target area can be improved by optimizing the placement of secondary loudspeakers, which can be viewed as a combinatorial optimization problem. Due to the non-convexity attribute of the problem, a genetic algorithm [9; 10; 11] and a simulated annealing algorithm [12; 13] have been applied to its study. However, the above-mentioned algorithms are prone to fall into local optimal solutions and the performance usually depends on the selection of parameters. Moreover, for more complex models, the parameter selection of the algorithms is more difficult and the computational time is exponentially higher [8]. An alternative way to approach this problem is to convert the non-convex optimization problem to a convex optimization problem by modifying the cost function. The main methods used in this approach are the least absolute shrinkage and selection operator (LASSO) [14] proposed by [PERSON] et al., the LASSO-LS method combining the LASSO and least-squares (LS) proposed by [PERSON] et al. [15; 16; 17], the constrained matching pursuit algorithm (CMP) [18], and the singular value decomposition algorithm (SVD) [19] proposed by [PERSON] et al. [PERSON] et al. compared these methods and concluded that the optimized placement of the loudspeaker using the CMP algorithm results in a better performance when the power constraint is considered, and a better performance using the LASSO algorithm when the power constraint is not considered [20]. [PERSON] et al. [21] proposed an iterative method based on acoustic contrast control (ACC) [22] algorithm to optimize the loudspeaker placement. Compared to the arc array, and the Gram-Schmidt orthogonalization (GSO) method [23], the iterative method has a better broadband performance. However, the above methods only consider the performance of the target area and neglect sound amplification in other areas. [PERSON] et al. [8] studied an algorithm combining the optimized secondary source array using the CMP algorithm and the zero sound pressure constraint points to control the forward-scattered sound of a rigid sphere that can improve the control performance in the forward-scattered sound target area and suppress the sound amplification in the other areas. However, this method does not control the backward-scattered sound from a rigid sphere. In reality, for incident waves impinging on rigid spheres or rigid cylinders, scattered acoustic energy is mostly concentrated in the area of the forward scattered sound (scattered sound with an angle between the direction of the scattered wave and the direction of the incident wave of \(0^{\circ}\) to \(90^{\circ}\)) and the area of the backward-scattered sound (scattered sound with an angle between the direction of the scattered wave and the direction of the incident wave of \(90^{\circ}\) to \(180^{\circ}\)) [8; 24].