To avoid any multi-hypothesis test and the complexity of some probability inequalities, the anomaly detection problem can be converted to the identification of a linear unknown parameter vector. Under priori condition about the number of the anomaly detection, a maximum likelihood identification problem and a non-convex sparse optimization problem are constructed. Then the optimum necessary condition is applied to solve the optimum estimation, and a solvable convex optimization is obtained from the non-convex sparse optimization by adopting a relaxation method. For different norm forms in the convex optimization, the optimum necessary condition and the fast gradient algorithm are respectively used to estimate the optimum values, and some convergence inequalities of the fast gradient algorithm are analyzed. Finally, the effectiveness of the proposed method is verified by the simulation example results.