= Solution
Along a fixed nonzero branch, $\|x(\mu)\|_2^2=1+\mu/\lambda_1\to1$ and the radial Hessian eigenvalue grows like $4\lambda_1^2/\mu$. The nonzero transverse eigenvalues $2(\lambda_i-\lambda_1)$ stay fixed. If $\lambda_1$ is simple, the ratio of the largest to smallest absolute eigenvalue therefore tends to infinity. If it repeats, the Hessian is already singular because of the zero transverse directions; there is no finite spectral condition number to begin with. These conclusions concern the nonzero branches: at the stationary origin $H=2(A+\mu I)$, and the same blowup is not implied.
Ill-conditioning does \b[not] invalidate barrier minimizers as approximations to the constrained solution. Let $\lambda_{\min}<0$. At radius $r$, the smallest possible quadratic value is $\lambda_{\min}r^2$, so global barrier minimization reduces to minimizing $\lambda_{\min}t-\mu\log(1-t)$ for $0\leq t<1$. For $0<\mu<-\lambda_{\min}$ its unique minimizing radius satisfies
$$
\boxed{r^2=1+\mu/\lambda_{\min},\qquad x(\mu)=r v_{\min},\quad\|v_{\min}\|_2=1.}
$$
Thus every limiting unit vector in that eigenspace solves the original problem, with objective $\lambda_{\min}$. For $\mu\geq-\lambda_{\min}$ the barrier minimizer is the origin instead. The increasing condition number concerns the numerical difficulty of small-parameter subproblems, which can call for scaling or a <primal-dual interior-point method>; it does not prevent the mathematical convergence just established. This is the <quadratic-ball barrier eigenvalue branch>.
\Image[/past-exam-of-the-numerical-linear-algebra-and-continuous-optimisation-course-of-the-university-of-oxford/2007-barrier-eigenvalue-branches.png]
{title=The nonzero barrier minimizers approach the unit sphere while their Hessian condition number grows; the origin remains stationary}
Back to article page
