Given a linear objective function and a system of linear constraints, is there any known closed form lower bounds for it? to clearly express the problem assume that $$ z(\mathbf{a,B,c})=\mathop {\inf} \left\{ {\bf{a^Tx} |\quad\bf{Bx-c}\le0} \right\} $$ is any closed form (rather linear) function $y(\mathbf{a,B,c})$ versus $\mathbf{a,B,c}$ such that $y(\mathbf{a,B,c})\le z(\mathbf{a,B,c})$.

As I know, any feasible dual solution for the above problem is a lower bound for it. However, the dual problem take the similar form of the primal problem. How can I obtain a close form lower bound for the above problem?