Interior (strictly feasible solution) for conic program

Question

For a conic program

$\min \ c^Tx$

$s.t. \ Ax+y\geq b,$

$\ \ \ \ \ \ \ \ x\in \mathcal{K}$

what a strictly feasible solution to this program?

Since $y\in \mathbb{R}^n$, I am not sure whether a solution $(x,y)$ with $x\in\text{int}(\mathcal{K})$ is a strictly feasible solution. Or there is no strictly feasible solution for this program? Thanks.

Answer 1

If you want a Slater point, $(x,y)$ with $x \in \mathrm{rel int}(\mathcal{K})$ and $Ax+y \geq b$ is sufficient (see page 226 in http://stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf ). The "rel" is relevant (consider, e.g., $\mathcal{K} = \{ (0,x) : x \in \mathbb{R} \}$).