Using approximate secant equations in limited memory methods for multilevel unconstrained optimization

The properties of multilevel optimization problems defined on a hierarchy of discretization grids can be used to define approximate secant equations, which describe the second‐order behavior of the objective function. Following earlier work by Gratton and Toint (2009) we introduce a quasi‐Newton method (with a linesearch) and a nonlinear conjugate gradient method that both take advantage of this new second‐order information. We then present numerical experiments with these methods and formulate recommendations for their practical use.