In this work, we propose a novel non-orthogonal pilot design framework to tackle the pilot contamination problem in multi-cell massive multiple input multiple output (MIMO) systems. Assuming the minimum mean square error (MMSE) estimators are adopted at the base stations (BSs), we design pilot signals under power constraints by minimizing the total mean square errors (MSEs) of the MMSE estimators of all BSs. The pilot design problem is difficult to solve due to the non-convex objective function. A linearized alternating direction method of multipliers (L-ADMM) algorithm is introduced to solve the above non-convex optimization problem. The L-ADMM algorithm could approximate the objective function in a linear form, which makes the original problem solvable. In addition, a new non-orthogonal pilot design problem that maximizes the total spectral efficiency of all users in the cellular network is established. We show that the proposed L-ADMM-based pilot design method can be directly extended to solve it. Finally, numerical simulations validate that the proposed pilot design framework can achieve higher channel estimation accuracy and uplink achievable sum rate compared to the state-of-the-art approaches with less computational complexity.