 Solution

For an active constraint with ${\lambda }_i^{\ast }>0$, the perturbed complementary slackness equation gives $c_i(x^k)\sim {\mu }_k/{\lambda }_i^{\ast }$. The Hessian matrix from part (c) therefore includesIf $\nabla c_i(x_{\ast })\ne 0$, 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.