I'm hoping to understand why interior-point methods require strict feasibility of inequality constraints in order to work correctly. From various texts that I've read, it is usually mentioned that IP methods get their name due to the strict feasibility requirement, but no explanation or proof of such a requirement is given.
2026-03-25 11:05:20.1774436720
Explanation of strict feasibility requirement for interior-point methods
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Basic interior-point methods would require that per definition as that is what the name interior-point algorithm refers to. These algorithms use a barrier involving (typically) $\log f(x)$ for $f(x)> 0$, and it should be clear that the algorithm collapses if $f(x)\leq 0$.
In practice though, many (most) interior-point methods aren't really interior-point, but have various penalty/infeasibility/homogenization/primal-dual techniques surrounding the basic interior-point framework to improve performance and make them more practical.