 Solution

For $p=-g\ne 0$, part (ii) simplifies to ${\alpha }_{\max }=2(1-\beta )/\gamma$. If the initial step is accepted, ${\alpha }_k={\alpha }_{\mathrm{init}}$. Otherwise the immediately preceding trial ${\alpha }_k/\tau$ failed and hence exceeded ${\alpha }_{\max }$. ThereforeThe fixed global Lipschitz constant and search parameters make this lower bound uniform across iterations. At $g=0$, the zero update passes the first trial and the sequence can be kept stationary.