Wolfe Conditions - Armijo Rule and Curvature

Armijo Rule and Curvature

Denote a univariate function restricted to the direction as . A step length is said to satisfy the Wolfe conditions if the following two inequalities hold:

i) ,

ii) ,

with . (In examining condition (ii), recall that to ensure that is a descent direction, we have .)

is usually chosen to quite small while is much larger; Nocedal gives example values of and for Newton or quasi-Newton methods and for the nonlinear conjugate gradient method. Inequality i) is known as the Armijo rule and ii) as the curvature condition; i) ensures that the step length decreases 'sufficiently', and ii) ensures that the slope has been reduced sufficiently.