I have an objective function containing the following
$$\pmb{\hat{C}_3(y)}= \sum_j \sum_{b} \sum_{w} \bigg[ d_{bj} \tau_{wbj} \bigg( y_{wbj1}\ \ (1-p(w)) + y_{wbj2} \sum_{w' \neq w} p(w') \ \ y_{{w'}bj1} \bigg) \bigg]$$
where $y \in \{0,1\}$. The rest are parameters. All the constraints that I have are linear.
Is there any way to relax this nonlinear problem other than introducing a new variable like $z = y \cdot y$? I am using LINGO to solve this problem and not even able to get a lower bound after many hours of running LINGO.
I don't think there is a way to do this without adding new decision variables. But, I think that's your best bet, or worth a shot anyway. Something like:
$$z_{ww'bj} = \begin{cases} 1, & \text{if }y_{wbj2} = y_{w'bj1} = 1 \\ 0, & \text{otherwise} \end{cases} $$
(I wasn't quite sure what the 1 and 2 at the end of the subscripts were so I basically ignored them.)
Then add constraints: $$\begin{align*} z_{ww'bj} & \ge y_{wbj2} + y_{w'bj1} - 1 \\ z_{ww'bj} & \le y_{wbj2} \\ z_{ww'bj} & \le y_{w'bj1} \end{align*}$$