Armijo criterion example

Question

Let $d_{k}$ be a descent direction for the function $f$ in $x_{k}\in\mathbb{R}^{n}$. Armijo criterion is $$ f(x_{k}+td_{k})\leq f(x_{k})+\alpha t\nabla f(x_{k})^{T}d_{k} $$

I have to find a function $f:\mathbb{R}\rightarrow\mathbb{R}$ for which every $t>0$ does not satisfy Armijo criterion when $\alpha=1$. Can anyone help me with an example of function please? Thanks.

Answer 1

Any strongly convex function satisfies, in particular, $$ f(x+td)>f(x)+t\nabla f(x)^Td, \quad t\ne 0,\ d\ne 0. $$ Hence, it cannot get $\le$ at the same time.

For example, $f(x)=\|x-a\|^2$, $x_k=0$, $d_k=a$, for any nonzero vector $a\in\Bbb{R}^n$ .