Consider the system of linear inequalities: $Ax \preceq b$.
The alternative system of inequalities is: $\lambda \succeq 0, A^T\lambda = 0, b^T \lambda \le 0$
These are strong alternatives since the optimum in the related system (A) below is achieved unless it is unbounded below.
(A) minimize $s$, subject to: $f_i(x) - s \le 0$ for $i=1, \dots, m; Ax=b$.
Can someone explain how these two systems are strong alternatives? I don't see how the unboundedness or optimal value of $A$ implies these two systems are strong alternatives.