 Cauchy point

For $g\ne 0$, the Cauchy point minimizes the quadratic model along the feasible part of the negative gradient ray. Writing $q=g^{T}Hg$, its step size isIndeed $m(-tg)=a-t\Vert g{\Vert }^2+\frac{1}{2}{t}^{2}q$ on $0\le t\le \Delta /\Vert g\Vert$. For $q\le 0$ it decreases throughout that interval; otherwise its unconstrained minimum occurs at $t=\Vert g{\Vert }^2/q$. The constant $a$ does not affect the minimizing step.