Trust-region acceptance ratio (${\rho }_k=\frac{f(x_k)-f(x_k+s_k)}{m_k(0)-m_k(s_k)}$)

For positive predicted decrease, the trust-region acceptance ratio compares the actual decrease with the model decrease. A ratio near one supports the local model; a negative ratio means the trial increased the objective function. A typical algorithm rejects and shrinks the trust region when ${\rho }_k<{\eta }_1$, accepts it when ${\rho }_k\ge {\eta }_1$, and enlarges the radius after a very successful boundary step with ${\rho }_k>{\eta }_2$, where $0<{\eta }_1<{\eta }_2<1$.