Conic dual problem

For ${\min }_x{c}^{\top }x$ subject to $Ax-b\in K$, the Lagrangian dual problem is ${\max }_y{b}^{\top }y$ subject to $A^{\top }y=c$ and $y\in K^{\ast }$. At feasible primal-dual points, the gap is $⟨Ax-b,y⟩\ge 0$. A self-dual cone has $K^{\ast }=K$, but self-duality alone does not imply feasibility or strong duality.