Abstract: The synthesis of mainlobe magnitude under sidelobe constraint is studied,which is a non-convex beampattern optimization problem including bilateral absolute value inequality.In view of the difficulty of traditional multi-constraint optimization algorithms to deal with non-convex structures,two iterative algorithms are proposed.One is to take a linear approximation to the original optimization problem,convert the non-convex constraint into an affine constraint,and then iterate the local second-order cone programming problem to solve the original problem.The other is to construct an augmented Lagrangian function by introducing auxiliary variables.The weighting vector is decoupled from constraints,and three optimization sub-problems about the original variables,mainlobe auxiliary variables and sidelobe auxiliary variables are solved alternately to obtain the solution of the initial non-convex problem.On the optimization sub-problems,low complexity solutions are constructed by flexible use of the Lagrange multiplier method.The complexity analysis of the two iterative algorithms is provided.The proposed algorithms are verified by both simulated and measured array manifold.The results show that both the two iterative algorithms can achieve the mainlobe magnitude approximation and synthesize the flat top beampattern,and have no dependence on the array manifold.The time consumption of the alternating iteration method is significantly lower than that of the iterative second-order cone programming method.