Past exam of the numerical linear algebra and continuous optimisation course of the University of Oxford / 2007 / B / 7 / ii / Solution

A Lipschitz gradient with constant $\gamma >0$ gives the descent lemmaFor $\alpha >0$, this upper bound is at most $f(x)+\alpha \beta g^{T}p$ whenever $\frac{\gamma }{2}\alpha \Vert p{\Vert }_2^2\le (\beta -1)g^{T}p$. Since $p$ is a descent direction, the right side is positive. Thus the Armijo condition holds throughoutAt $\alpha =0$ it holds by equality. If the gradient Lipschitz constant can be taken as zero, the objective is affine and every positive step along a descent direction passes this sufficient-decrease test.