duality theorem for homogeneous system

Question

hi there what is dual of a homogeneous system?

Primal: $$\text{Max}\ cx $$

$$ Ax \le 0 $$

Dual : $$Min\ 0 $$

$$A^{T}y=c$$

$$y\ge 0$$

but what does it mean $Min\ 0$ in dual problem?

Answer 1

It means it is a feasibility problem.

Any feasible solution would make the objective function $0$, which is optimal.

If the dual is feasible, then clearly, $x=0$ is also an optimal solution to the primal by strong duality.