Wolfe Conditions

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 f:\mathbb R^n\to\mathbb R be a smooth objective function, and let \mathbf{p}_k be a given search direction. A step length \alpha_k is said to satisfy the Wolfe conditions if the following two inequalities hold.
i) f(\mathbf{x}_k+\alpha_k\mathbf{p}_k)\leq f(\mathbf{x}_k)+c_1\alpha_k\mathbf{p}_k^{\mathrm T}\nabla f(\mathbf{x}_k),
ii) \mathbf{p}_k^{\mathrm T}\nabla f(\mathbf{x}_k+\alpha_k\mathbf{p}_k)\geq c_2\mathbf{p}_k^{\mathrm T}\nabla f(\mathbf{x}_k),
with 0. Inequality i) is known as the Armijo condition and ii) as the curvature condition. i) demands that \alpha_k gives 'sufficient' decrease in f, and ii) ensures that the slope of the function \phi(\alpha)=f(\mathbf{x}_k+\alpha\mathbf{p}_k) at \alpha_k is greater than c_2 times that at 0. The Wolfe conditions provide a computationally more attractive way of computing a step length than minimizing exactly \phi over \alpha\in\mathbb R. However, the conditions can result in a value for the step length that is not close to a minimizer of \phi. If we modify the curvature condition to say
iia) |\mathbf{p}_k^{\mathrm T}\nabla f(\mathbf{x}_k+\alpha_k\mathbf{p}_k)|\leq c_2|\mathbf{p}_k^{\mathrm T}\nabla f(\mathbf{x}_k)|
then i) and iia) together are the so-called strong Wolfe conditions, and as such \alpha_k is forced to lie close to a critical point of \phi.

Reference

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

 

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