2019/04/09 by Serge Gratton, Calandra, Henri, Elisa Riccietti +4 · 2 citations
Computer Science · Mathematics · #Advanced Optimization Algorithms Research #FOS: Mathematics #Iterative Methods for Nonlinear Equations #Numerical Analysis (math.NA) #Stochastic Gradient Optimization Techniques
paper · doi:10.48550/arxiv.1904.04692
openalex publication_date 2019/04/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We propose a new family of multilevel methods for unconstrained minimization. The resulting strategies are multilevel extensions of high-order optimization methods based on q-order Taylor models (with q >= 1) that have been recently proposed in the literature. The use of high-order models, while decreasing the worst-case complexity bound, makes these methods computationally more expensive. Hence, to counteract this effect, we propose a multilevel strategy that exploits a hierarchy of problems of decreasing dimension, still approximating the original one, to reduce the global cost of the step computation. A theoretical analysis of the family of methods is proposed. Specifically, local and global convergence results are proved and a complexity bound to reach first order stationary points is also derived. A multilevel version of the well known adaptive method based on cubic regularization (ARC, corresponding to q = 2 in our setting) has been implemented. Numerical experiments clearly highlight the relevance of the new multilevel approach leading to considerable computational savings in terms of floating point operations compared to the classical one-level strategy.