Moving target detection and tracking in reverberation environment is an important yet challenging problem in many applications such as speech, sonar, radar and seismic signal processing. Extending the early work of online subspace and sparse filtering [1], this paper presents an approach based on structured convex optimization. Exploiting potentially coherent structure of reverberation background, we represent the beamspace image data as the sum of a low-rank and a sparse matrix, where reverberation assumes low-rank structure and moving target signal is modeled as sparse. Detection and tracking is then formulated as a structured convex optimization problem, and solved via an accelerated proximal gradient (APG) algorithm. The performance of proposed algorithm is demonstrated using experimental results.