Past exam of the numerical linear algebra and continuous optimisation course of the University of Oxford / 2007 / B / 8 / iii

The logarithmic barrier isIts gradient is $2Ax+2\mu x/(1-x^{T}x)$. Thus every nonzero stationary point satisfiesA fixed negative-eigenvalue branch exists for $0<\mu <-{\lambda }_1$. The zero point is stationary for every $\mu$ as well, and it is not an eigenvector; at zero the displayed scalar $-\mu$ need not be an eigenvalue of $A$. Thus the claimed eigenvalue conclusion requires the nonzero-point qualification. This distinction is essential for the following Hessian analysis.