= Ill-conditioning of a logarithmic barrier
Near an <active constraint> with limiting <Lagrange multiplier> $\lambda_i^*>0$, a convergent barrier sequence has $c_i(x_k)\sim\mu_k/\lambda_i^*$. Its <Hessian matrix> therefore contains the term
$$
\frac{\mu_k}{c_i(x_k)^2}\nabla c_i(x_k)\nabla c_i(x_k)^T\sim\frac{(\lambda_i^*)^2}{\mu_k}\nabla c_i(x_*)\nabla c_i(x_*)^T.
$$
For a nonzero constraint <gradient>, this gives large curvature in a direction normal to the boundary. If a nonzero common tangent direction retains positive curvature of order one, while the remaining terms stay bounded, the <condition number> typically grows like $1/\mu_k$. This is a mechanism rather than a universal rate: there may be no tangent direction, all curvatures may grow together, or an <active constraint> may have zero limiting <Lagrange multiplier>.
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