Conic phase-I problem

An auxiliary conic optimization problem searches for strict feasibility before following a central path. Given $e\in \mathrm{int}K$, minimizing $t$ subject to $Ax-b+te\in K$ and $t\ge -1$ has a strict feasible start for sufficiently large $t$. Any feasible solution with $t<0$ certifies $Ax-b\in \mathrm{int}K$. If the original problem is strictly feasible, a small negative $t$ is feasible. Equality constraints and dual feasibility require their corresponding auxiliary procedures.