A step length is said to satisfy the Wolfe conditions, restricted to the direction , if the following two inequalities hold:
with . (In examining condition (ii), recall that to ensure that is a descent direction, we have , as in the case of gradient descent, where , or Newton–Raphson, where with positive definite.)
is usually chosen to be quite small while is much larger; Nocedal and Wright give example values of and for Newton or quasi-Newton methods and for the nonlinear conjugate gradient method.[3] Inequality i) is known as the Armijo rule[4] and ii) as the curvature condition; i) ensures that the step length decreases 'sufficiently', and ii) ensures that the slope has been reduced sufficiently. Conditions i) and ii) can be interpreted as respectively providing an upper and lower bound on the admissible step length values.