Global convergence using de-linked Goldstein or Wolfe linesearch conditions

Goldstein or Wolfe conditions can be imposed on a linesearch to ensure convergence of an iterative nonlinear optimization algorithm to a stationary point. However it is actually not necessary to find a single step which satisfies both Goldstein (or both Wolfe) conditions simultaneously in order to ensure global convergence. De-linking the conditions can make it significantly easier to find an acceptable stepsize, which is neither too short nor too long. Although this fact has been known for a long time, the practice seems to have fallen out of fashion. However In this note we give a short, self-contained proof of global convergence for de-linked Goldstein and Wolfe conditions, and advocate their use. In particular, we argue that the increasingly widespread availability of second order adjoints via Automatic Differentiation tools means that the cost of a conventional safe line search is often unacceptably high for algorithms such as Truncated Newton. The de-linked approach advocated here is used with the Goldstein conditions in the OPTIMA Truncated Newton code.