A sparse nonlinear inverse scattering problem arising in microwave imaging is analyzed and numerically solved for retrieving dielectric contrast of region of interest from measured fields. The proposed approach is motivated by a Tikhonov functional incorporating a sparsity promoting $l_{1}$-penalty term. The proposed iterative algorithm of soft shrinkage type enforces the sparsity constraint at each nonlinear iteration and provides an effective reconstructions of unknown (complex) dielectric profiles. The scheme produces sharp and good reconstruction of dielectric profiles in sparse domains and keeps its convergence during the reconstruction. Numerical results present the effectiveness and accuracy of the proposed method by imposing positivity and smoothness constraints.