= Cauchy decrease bound
{c}
{title2=$m(0)-m(s_C)\geq\tfrac12\|g\|\min\{\Delta,\|g\|/\|H\|\}$}
Use the induced <spectral norm> for the <symmetric matrix> $H$ and interpret $\|g\|/\|H\|=+\infty$ when $H=0$. If $q=g^THg\leq0$, the <Cauchy point> gives a decrease at least $\Delta\|g\|$. If $q>0$, its <step size> satisfies $\tau q\leq\|g\|^2$ and $q\leq\|H\|\|g\|^2$, so
$$
m(0)-m(s_C)=\tau\|g\|^2-\tfrac12\tau^2q\geq\tfrac12\tau\|g\|^2\geq\tfrac12\|g\|\min\{\Delta,\|g\|/\|H\|\}.
$$
Thus the numerical constant $1/2$ is independent of the model data. The bound links a nonzero <gradient> to a definite predicted decrease, including for an <indefinite matrix> $H$.
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