Ill-conditioning of a logarithmic barrier

Near an active constraint with limiting Lagrange multiplier ${\lambda }_i^{\ast }>0$, a convergent barrier sequence has $c_i(x_k)\sim {\mu }_k/{\lambda }_i^{\ast }$. Its Hessian matrix therefore contains the termFor 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.