= Solution
For an <active constraint> with $\lambda_i^*>0$, the perturbed <complementary slackness> equation gives $c_i(x^k)\sim\mu_k/\lambda_i^*$. The <Hessian matrix> from part (c) therefore includes
$$
\frac{\mu_k}{c_i(x^k)^2}\nabla c_i(x^k)\nabla c_i(x^k)^T\sim\frac{(\lambda_i^*)^2}{\mu_k}\nabla c_i(x_*)\nabla c_i(x_*)^T.
$$
If $\nabla c_i(x_*)\ne0$, curvature normal to that boundary grows like $1/\mu_k$. By contrast, a direction tangent to every <active constraint> annihilates these large normal <outer products> at the limiting point, and its curvature may remain of order one. A large spread of <eigenvalues> then makes the Newton linear system poorly conditioned. Under bounded remaining curvature and a positive bounded tangential curvature, the <condition number> typically grows like $1/\mu_k$; this is the usual <ill-conditioning of a logarithmic barrier>.
It is not a universal rate: there may be no nonzero common tangent direction, some limiting <Lagrange multipliers> may vanish, or all curvatures may grow together. The <Hessian> formula identifies the source of the difficulty without assuming these extra nondegeneracy conditions.
Back to article page
