In (unconstrained) optimization, the Wolfe conditions are a set of inequalities for performing inexact linesearches; that is, for efficiently selecting a step length in the linesearch algorithm.

Let be a smooth objective function, and let be a given search direction. A step length α_k is said to satisfy the Wolfe conditions if the following two inequalities hold.

i) ,

ii) ,

with 0 < c₁ < c₂ < 1. Inequality i) is known as the Armijo condition and ii) as the curvature condition. i) demands that α_k gives 'sufficient' decrease in f, and ii) ensures that the slope of the function at α_k is greater than c₂ times that at 0.

The Wolfe conditions provide a computationally more attractive way of computing a step length than minimizing exactly φ over . However, the conditions can result in a value for the step length that is not close to a minimizer of φ. If we modify the curvature condition to say

iia)

then i) and iia) together are the so-called strong Wolfe conditions, and as such α_k is forced to lie close to a critical point of φ.

Reference

J. Nocedal and S. J. Wright, Numerical optimization. Springer Verlag, New York, NY, 1999.