Hi I am working on the following question:
If $c \in int(N_P(x))$, then $x$ is a unique solution.
I have proven that this is true if $x$ is a vertex. Well I am wondering if the following is a counter example: Let $Q=[-1,1]^2$. Choose $x=(1,0)$. Then $N_Q(x)=pos((1,0))$ where $pos$ is the positive hull. Since $aff(x)=lin((1,0)^T)$ ($aff,lin$ the affine respectively linear space) $c=(1,0) \in int(N_Q(x))$, but $x'=(1,0.5)$ is also an optimal solution.
Consider the LP max $c^Tx,x \in P$, where $c \in \mathbb{R^n}$ and $P= \{x \in \mathbb{R^n}: Ax \leq b \}$ is a fulldimensional polyhedron with $A\in \mathbb{R^{m\times n}}$ and $b\in \mathbb{R^m}$ Moreover let $x \in P$.
If $c \in int(N_P(x))$ then there exists a unique dual optimal solution. True or False?
Probably not if the primal has no unique solution the dual won't.
Is it possible to conlcude the primal has a unique solution if and only if the dual has?