The convex sparse penalty based compressive beamforming technique can achieve robust high resolution in direction-of-arrival (DOA) estimation tasks, but it often leads to an insufficient sparsity-inducing problem due to its convex loose approximation to ideal ℓ0 nonconvex penalty. On the contrary, the nonconvex sparse penalty can tightly approximate ℓ0 penalty to effectively enhance DOA estimation accuracy, but it incurs an initialization sensitivity problem due to its multiple local minimas. Leveraging their individual advantages, a minimax-concave penalty (MCP) regularized DOA estimation algorithm is proposed to achieve a maximally sparse level while maintaining the convex property of the overall objective function. Moreover, an accelerated block gradient descent-ascent algorithm with convergence guarantee is developed to rapidly achieve its one optimal point. Simulation results demonstrate that MCP penalty improves DOA estimation accuracy compared with popular sparse compressive beamforming techniques in strong noise scenarios and weak source confirmation. Ocean experimental results also validate that it retains more stable DOA estimation accuracy and incurs less artificial interferences.