Adaptive scheme for bracketing-based Wolfe line searches

Determine whether an adaptive line-search scheme analogous to the proposed adaptive backtracking—one that leverages the degree of violation of the line-search criterion to set step-size factors—can be constructed for bracketing-based procedures that enforce the weak and strong Wolfe conditions, which are not affine in the step size.

Background

The paper introduces adaptive backtracking factors that depend on how much a line-search criterion (e.g., Armijo or the descent lemma) is violated, yielding more aggressive yet provably safe step-size adjustments for criteria that are affine in the step size.

Weak and strong Wolfe conditions are standard in line-search routines for methods such as quasi-Newton, but they are not affine in the step size and are typically enforced via bracketing rather than backtracking. The authors explicitly state it is unclear whether an analogous adaptive scheme exists in this bracketing context, motivating the problem of determining the existence and design of such a scheme.