How to show that the following property is true?

Question

If $p_1^*(a)=\sup\{-b^Tz_1| A^Tz_1+a=0, z_1\succcurlyeq 0\}>\gamma$ then how to show that this is equivalent to $A^Tz_1+a=0, z_1\succcurlyeq 0, b^Tz_1 < \gamma$. Any help in this regard will be much appreciated. $A,a, b, \gamma$ are constants in these expressions and $z_1$ is variable. (I found this in the solution manual of Convex optimization (by Boyd and Vandenberghe) book problem 5.18).

Answer 1

The solution manual has an error. If $$\begin{pmatrix}A \\ a^T\end{pmatrix}x \leq \begin{pmatrix}b \\ \gamma\end{pmatrix}$$ has a solution, then Farkas lemma (third variant on Wikipedia) states that: $$A^Tz_1+aw_1=0, b^Tz_1+\gamma w_1<0, z_1\geq 0, w_1\geq 0$$ has no solution. Note the + in front of $\gamma$, which explains the difference in sign.