In this lecture note, the author introduces Simplex method in Section 4 (page 20) by setting up the following problem:
Find $y$ to minimize $y^Tb$ subject to $y\geq0$ and $y^TA\geq c^T$.
In the 3rd paragraph of the same page the author then writes:
"if $-c\geq0$ and $b\geq0$, there is an obvious solution to the problem; namely, the minimum occurs at $y=0$ ..."
The next sentence he writes a succinct explanation but I still do not get it. He introduces a slack variable in the same page, which I think should aid our understanding of the statement above, but it doesn't help. How would you rephrase the explanation?
Without the slack variable, taking $y=0$ satisfies both constraints: $y \ge 0$ and $y^T A = 0\ge c^T$ (because $-c \ge 0$, equivalently, $c \le 0$). So $y=0$ is feasible. Now $b \ge 0$ and $y \ge 0$ imply that $y^Tb \ge 0$. Because $y=0$ attains this lower bound, it is optimal.
With the slack variable, taking $(y,s)=(0,-c)$ satisfies constraints $y \ge 0$, $s \ge 0$ (because $-c \ge 0$), and $s^T = -c^T = 0^T A - c^T = y^T A - c^T$. So $(y,s)=(0,-c)$ is feasible. Now $b \ge 0$ and $y \ge 0$ imply that $y^T b \ge 0$. Because $(y,s)=(0,-c)$ attains this lower bound, it is optimal.