The Lagrangian relaxation approach is used to generate lower (upper) bounds for minimization (maximization) problems by moving some constraints to the objective function and multiplying them by "Lagrangian multipliers".
The sub-gradient algorithm tries to improve the bounds by updating the Lagrangian multipliers. There are some codes which initialize the Lagrangian multipliers by the dual value of the constraints in the optimal solution of the linear relaxation of the problem (e.g., https://www.gams.com/latest/gamslib_ml/libhtml/gamslib_gapmin.html).
This approach is problem-independent and seems to perform better than initialization by zero vectors. However, I could not find any textbook or paper which suggested or used this initialization. I wondered if you could help by introducing such a reference.