Fraction of Cauchy decrease

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.