= Cauchy point
{c}
{title2=$s_C=-\tau g$}
For $g\ne0$, the <Cauchy point> minimizes the quadratic model along the feasible part of the negative <gradient> ray. Writing $q=g^THg$, its <step size> is
$$
\tau=\begin{cases}\Delta/\|g\|,&q\leq0,\\\min\{\Delta/\|g\|,\|g\|^2/q\},&q>0.\end{cases}
$$
Indeed $m(-tg)=a-t\|g\|^2+\tfrac12t^2q$ on $0\leq t\leq\Delta/\|g\|$. For $q\leq0$ it decreases throughout that interval; otherwise its unconstrained <minimum> occurs at $t=\|g\|^2/q$. The constant $a$ does not affect the minimizing step.
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