Let's go to the general problem of having lower bounds. There are many ways to get lower bounds to the problems. The typical one is by solving another problem called the relaxation of the original problem. Relaxation means that we have another function as an objective function and another set and we pretend that whenever a point is within the original set then the new function must be a lower bound for the original function. This way we always will have a lower bound by solving the relaxation.