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Let $K\subset \mathbb{R}^2$ be a closed convex and pointed cone, $A$ be a $2\times 2$ square matrix and $b, c\in \mathbb{R}^2$. Consider the problem $$ (P)\quad \min\{\langle c, x\rangle: Ax\geq_K b\}, $$ and its dual $$ (D)\quad \max\{\langle b, y\rangle: A^Ty=c,\; y\in K^*\}, $$ where $$ K^*=\{y\in\mathbb{R}^2: \langle x, y\rangle\geq 0, \forall x\in K\}. $$ I would like to construct $(P)$ and $(D)$ such that $$ \inf(P)-\sup(D)>0. $$ Thank you for all comments and helping.

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