Cauchy decrease bound ($m(0)-m(s_C)\ge \frac{1}{2}\Vert g\Vert \min \{\Delta ,\Vert g\Vert /\Vert H\Vert \}$)

Use the induced spectral norm for the symmetric matrix $H$ and interpret $\Vert g\Vert /\Vert H\Vert =+\infty$ when $H=0$. If $q=g^{T}Hg\le 0$, the Cauchy point gives a decrease at least $\Delta \Vert g\Vert$. If $q>0$, its step size satisfies $\tau q\le \Vert g{\Vert }^2$ and $q\le \Vert H\Vert \Vert g{\Vert }^2$, soThus 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 $H$.

A feasible trial step $s_k$ achieves a fixed Fraction of Cauchy decrease if $m_k(0)-m_k(s_k)\ge \kappa [m_k(0)-m_k(s_{C,k})]$ for one constant $\kappa \in (0,1]$ independent of $k$. The Cauchy decrease bound then supplies a quantitative model decrease without solving the trust-region subproblem exactly. Both the fraction and the relevant model regularity must be controlled in a convergence argument.