Yes! there is one of this type. Any numerical issues with them?

Anyway, its important to tell that even solving the subproblem up to optimality I face this issue of the bigger bound

You are right. But isn't that true only for the lag. function upper bound? Because If its optimal value is always lower than the original problem optimal objective, the lag. function lower bound would be a valid lower bound for the original problem. Not sure if I'm abusing the subgradient theory here with this simplification...

Which bound? The relaxed model bound is very far... The LR best bound was actually good for smaller instances. For larger ones, as a proof of concept, I was solving with time limit, and to avoid invalid bounds I was using the Lagrangian problem lower bound just to check the algorithm flow faster

With the original MIP after some long runtime. Now I'm using this value as an initial upper bound for the subgradient