= Newton minimization of the square-root norm
{c}
{title2=$f(z)=\sqrt{1+\|z\|^2},\quad z_{k+1}=-\|z_k\|^2z_k$}
For the <Euclidean norm> on $\mathbb R^d$, the regularized radial <function> $f(z)=\sqrt{1+\|z\|^2}$ has <gradient> $z/f$ and <Hessian> $I/f-zz^T/f^3$. The <Cauchy-Schwarz inequality> makes its <Hessian> a <positive-definite matrix>, so $f$ is strictly <convex> with unique <minimum> at zero. Its <inverse matrix> is $f(I+zz^T)$: multiplication verifies this using $(zz^T)^2=\|z\|^2zz^T$ and $f^2=1+\|z\|^2$. Substitution in <Newton's method in optimization> therefore gives $z_{k+1}=-\|z_k\|^2z_k$. Consequently $\|z_{k+1}\|=\|z_k\|^3$: nonzero starts in the open unit ball have exact <cubic convergence>, starts on the unit sphere alternate between opposite points, and starts outside it diverge. This example separates strict <convexity> from global <convergence> of an undamped iteration.
Back to article page
