Consider the primal problem $ min ct x $ s.a $ Ax = b, x \geq 0, $ where $ A \in \mathbb {R}^{ m × n}$ has put full line. Suppose that, in a certain iteration of the dual-simplex method, relative to a dual-feasible base $A_B \in \mathbb {R}^{m × n}$, for some $ r \in$ ${1,. . . , m}$ happens $b_{r} <0 $ and $ a_{rs} \geq 0$ for every $ j = 1,..,n.$ Determine a direction of recession of the dual that is from growth to dual function.
Bazaraa have a help, he said to consider $w=c_B.A^{-1}_B + B^r$, where $B^r$ is a r row of $A^{-1}_B$. But i can't see how this can be true. Anyway, i would like to proof that w is in fact a recession ray, in other words, $Aw \leq 0$, $w\geq 0$ and $w \neq 0$
Thanks.