Let $ \alpha $ and $ \beta $ be 2 scalars in $ [0,1]
$. Let $ x $ be the current value of the unknown, $ d
$ the search direction and $ t $ the step. Let $ f $
be the function to minimize. The line search stops when $ t
$ verifies
$$ f(x + t \cdot d) - f(x) \leq -\alpha t f'(x+t \cdot d) $$
and
$$ f(x+\frac{t}{\beta} \cdot d) - f(x) > -\frac{\alpha}{\beta}
t f'(x+t \cdot d) $$
(see Polak, Algorithms and consistent approximations, Optimization,
volume 124 of Applied Mathematical Sciences, Springer-Verlag, NY,
1997)
Armijo line search.
Let $ \alpha $ and $ \beta $ be 2 scalars in $ [0,1] $. Let $ x $ be the current value of the unknown, $ d $ the search direction and $ t $ the step. Let $ f $ be the function to minimize. The line search stops when $ t $ verifies $$ f(x + t \cdot d) - f(x) \leq -\alpha t f'(x+t \cdot d) $$ and $$ f(x+\frac{t}{\beta} \cdot d) - f(x) > -\frac{\alpha}{\beta} t f'(x+t \cdot d) $$
(see Polak, Algorithms and consistent approximations, Optimization, volume 124 of Applied Mathematical Sciences, Springer-Verlag, NY, 1997)