Wolfe Conditions - Strong Wolfe Condition On Curvature

Strong Wolfe Condition On Curvature

The Wolfe conditions, however, can result in a value for the step length that is not close to a minimizer of . If we modify the curvature condition to the following,

iia)

then i) and iia) together form the so-called strong Wolfe conditions, and force to lie close to a critical point of .

The principal reason for imposing the Wolfe conditions in an optimization algorithm where is to ensure convergence of the gradient to zero. In particular, if the cosine of the angle between and the gradient,

is bounded away from zero and the i) and ii) hold, then .

An additional motivation, in the case of a quasi-Newton method is that if, where the matrix is updated by the BFGS or DFP formula, then if is positive definite ii) implies is also positive definite.