Proving statement based on given known optimal value properties

Question

If we have (P): $\max\{c^Tx:Ax=b, \space x\geq \overrightarrow0\}$

Then (D) is: $\min\{b^Ty: A^Ty\geq c \}$

Let $n$ be the number of rows of $A^T$.

Prove if for all $i\in\{1,\dots,n\}$, there is an optimal solution for (P), $x^i$, such that $x^i_i=0$ then $b=\overrightarrow0$.

Answer 1

The Claim is wrong .

Take problem (P) as $\max\{0x+0y ~: ~ x+y=1, \space x ,y\geq \overrightarrow0\}$

Then $x_1 = (0,1)$ and $x_2 = (1,0)$ are two optimal solutions, but $b=1 \neq 0$