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.
Read more about this topic: Wolfe Conditions
Famous quotes containing the word rule:
“To me the “female principle” is, or at least historically has been, basically anarchic. It values order without constraint, rule by custom not by force. It has been the male who enforces order, who constructs power structures, who makes, enforces, and breaks laws.”
—Ursula K. Le Guin (b. 1929)