We considerer the following optimization problem $$ \left\{\begin{array}{cl} \max\limits_{x\in\mathcal{C}} & f(x) \\[2pt] \text{s.t.} & \mathcal{A}x-b \in K \end{array} \right. \tag{1} $$
where $f:\mathbb{R}^{n}\rightarrow \mathbb{R}$ and $\mathcal{A}:\mathbb{R}^{n}\rightarrow \mathbb{R}^{m}$ are linear functions non-zero, $b\in\mathbb{R}^{m}$, $\mathcal{C}$ is a convex cone in $\mathbb{R}^{n}$ and $K$ is a closed convex cone in $\mathbb{R}^{m}$.
Question: I need to find conditions over a set $U\subset\mathbb{R}^{n}$ such that if $\mathcal{C}$ is the convex cone generated by $U$, then problem $(1)$ is equivalent to the following problem $$ \left\{\begin{array}{cl} \max\limits_{x\in U} & f(x) \\[2pt] \text{s.t.} & \mathcal{A}x-b \in K \end{array} \right. \tag{2} ?$$