I am trying to solve the following problem. The decisions in the problem are $x, y, v, $ and $W$, where $x, y$ are binary and $v, W$ are continuous variables.
\begin{equation}\label{eq:3} \begin{aligned} & \underset{x, y, v, W}{\text{minimize}} & & c^{T}v + W^{T}\mathbf{1}\\ & \text{subject to} & & Ax \le b\\ & & & \sum_{f = 1}^{q} y_{lf} = x_l, \forall l = 1, ..., p\\ & & & Hv = d\\ & & & v_a \le \left(\sum_{f = 1}^{q} f y_{l(a)f}\right) W_i, \forall a = 1, ..., m, \forall i = 1, .., n \\ & & &x \in \{0, 1\}^p, y \in \{0, 1\}^{p\times q}, v \in \mathbb{R}^{m}_{+}, W \in \mathbb{R}^{n}_{+} \end{aligned} \end{equation}
As we can see that there is a product of continuous and binary variable in the fourth set constraints, so I linearize that using McCormick Relaxation. Assuming $0 \le W_i \le \bar{W}_{i}$ and let $t_{fai} = y_{l(a)f} \times W_i$, we can state the problem as an exact MIP as below:
\begin{equation} \begin{aligned} & \underset{x, y, v, W, t}{\text{minimize}} & & c^{T}v + W^{T}\mathbf{1}\\ & \text{subject to} & & Ax \le b\\ & & & \sum_{f = 1}^{q} y_{lf} = x_l, \forall l = 1, ..., p\\ & & & Hv = d\\ & & & v_a \le \left(\sum_{f = 1}^{q} f t_{fai} \right), \forall a = 1, ..., m, \forall i = 1, .., n \\ & & & W_i - t_{fai} \le \bar{W}_{i} (1 - y_{l(a)f}), \forall f = 1, ..., q, \forall a = 1, ..., m, \forall i = 1, .., n\\ & & & t_{fai} \le \bar{W}_{i} y_{l(a)f}, \forall f = 1, ..., q, \forall a = 1, ..., m, \forall i = 1, .., n\\ & & & W_i - t_{fai} \ge 0, \forall f = 1, ..., q, \forall a = 1, ..., m, \forall i = 1, .., n\\ & & & t_{fai} \ge 0, \forall f = 1, ..., q, \forall a = 1, ..., m, \forall i = 1, .., n\\ & & &x \in \{0, 1\}^p, y \in \{0, 1\}^{p\times q}, v \in \mathbb{R}^{m}_{+}, W \in \mathbb{R}^{n}_{+} \end{aligned} \end{equation}
I am applying Benders Decomposition to solve the problem. The complicating variables are $x$ and $y$. Due to a large value of $\bar{W}_{i}$, the lower bound to the problem is really bad. The iterations look as below. The algorithm does not converge and becomes slow after adding lots of optimality cuts. I tried different values of $\bar{W}_i$. Letting $\bar{W}_i$ too low makes the subproblem infeasible (but my subproblem has been proved to have feasible solution for any value of $x, y$ so it should not gernerate any feasibility cuts). Are there any possible solutions to avoid this problem? I appreciate any help.
(it = 1, LB = -1249013856, UB = 780802)
(it = 2, LB = -260761666, UB = 647047)
(it = 3, LB = -260385041, UB = 780652)
(it = 4, LB = -255998781, UB = 729034)
(it = 5, LB = -255661092, UB = 765904)