I have a two-stage stochastic program as follows: \begin{align}\max&\quad f^\top y+\sum_{s}p_sc_s^\top x_s\\\text{s.t.}&\quad Ay=b\\&\quad W_sX_s+Ty \le h_s \quad \forall s \in S \\&\quad P_sx_s \le q_s \quad \forall s \in S \\&\quad x_s \ge 0\\&\quad y\in \{0,1\}&\quad\end{align} and I want to solve it with Lagrangian relaxation. I relaxed the constraints $$W_sX_s+Ty \le h_s \quad \forall s \in S,$$ and it is decomposed to a first-stage problem and $|S|$ sub-problems related to the second stage. I solved the problem but there is a huge gap between lower and upper bounds. I think I correctly applied the algorithm (Gurobi+Python), but I do not know why it happens.
Also, I have used Benders decomposition to solve this problem and because the coupling constraints are related to capacity constraints and because these constraints are not active in the optimal solution, the dual variables corresponding to these constraints are mostly zero and in generated cuts, most $y$ variables have coefficients $0$ and it does not converge (or at least in a reasonable number of iterations).
I also used combinatorial Benders decomposition but it just excludes each solution and works like enumeration.
I would be thankful if someone has a similar experience and let me know what I can do.