On High-Order Multilevel Optimization Strategies

We propose a new family of multilevel methods for unconstrained minimization. resulting strategies are multilevel extensions of high-order optimization based on q-order Taylor models (with q >= 1) that have been recently in the literature. The use of high-order models, while decreasing the-case complexity bound, makes these methods computationally more. Hence, to counteract this effect, we propose a multilevel strategy exploits a hierarchy of problems of decreasing dimension, still the original one, to reduce the global cost of the step. A theoretical analysis of the family of methods is proposed., local and global convergence results are proved and a complexity to reach first order stationary points is also derived. A multilevel of the well known adaptive method based on cubic regularization (ARC, to q = 2 in our setting) has been implemented. Numerical clearly highlight the relevance of the new multilevel approach to considerable computational savings in terms of floating point compared to the classical one-level strategy.