The paper presents a sparsity-driven method utilizing loudspeakers to reconstruct spatial sound fields using measurements obtained from a spherical microphone array (SMA). Employing spherical harmonics decomposition (SHD), the SMA recordings are characterized in the spherical harmonics domain. The gains for the loudspeakers are determined through an optimization problem, equating spherical harmonics pressure coefficients from primary and secondary sources. Furthermore, the sparsity within the loudspeaker feeds is redefined as a constrained sparse optimization problem, integrating linearity and orthogonality constraints. This method effectively reduces the required loudspeakers while maintaining sound field quality. The Bregman iteration method is applied to solve the constrained optimization problem. Rigorous evaluation based on reconstructed sound fields and objective measures highlights significant enhancements compared to least square and compressed sensing methods.