Existing LiDAR-inertial state estimation assumes that the state at the beginning of current sweep is identical to the state at the end of last sweep. However, if the state at the end of last sweep is not accurate, the current state cannot satisfy the constraints from LiDAR and IMU consistently, ultimately resulting in local inconsistency of solved state (e.g., zigzag trajectory or high-frequency oscillating velocity). This paper proposes a semi-elastic optimization-based LiDAR-inertial state estimation method, which imparts sufficient elasticity to the state to allow it be optimized to the correct value. This approach can preferably ensure the accuracy, consistency, and robustness of state estimation. We incorporate the proposed LiDAR-inertial state estimation method into an optimization-based LiDAR-inertial odometry (LIO) framework. Experimental results on four public datasets demonstrate that: 1) our method outperforms existing state-of-the-art LiDAR-inertial odometry systems in terms of accuracy; 2) semi-elastic optimization-based LiDAR-inertial state estimation can better ensure consistency and robustness than traditional and elastic optimization-based LiDAR-inertial state estimation. We have released the source code of this work for the development of the community.