Why does an LP with empty interior feasible space (a lower-dimensional polytope) cause standard interior-point methods to fail or require special regularization?
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Standard interior-point methods rely on strict interior feasibility ($x > 0$) to compute logarithmic barrier functions. If the feasible region is lower-dimensional, variables are forced to zero everywhere, causing the barrier to evaluate to infinity, the KKT matrix to become singular, and the central path to collapse.
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