Scalable adaptive cubic regularization methods

daptive cubic regularization (ARC) methods for unconstrained optimization compute steps from linear systems involving a shifted Hessian in the spirit of the Levenberg-Marquardt and trust-region methods. The standard approach consists in performing an iterative search for the shift akin to solving the secular equation in trust-region methods. Such search requires computing the Cholesky factorization of a tentative shifted Hessian at each iteration, which limits the size of problems that can be reasonably considered. We propose a scalable implementation of ARC named ARC _q K in which we solve a set of shifted systems concurrently by way of an appropriate modification of the Lanczos formulation of the conjugate gradient (CG) method. At each iteration of ARC _q K to solve a problem with n variables, a range of m ≪ n shift parameters is selected. The computational overhead in CG beyond the Lanczos process is thirteen scalar operations to update five vectors of length m , and two n -vector updates for each value of the shift. The CG variant only requires one Hessian-vector product and one dot product per iteration, independently of m . Solves corresponding to inadequate shift parameters are interrupted early. All shifted systems are solved inexactly. Such modest cost makes our implementation scalable and appropriate for large-scale problems. We provide a new analysis of the inexact ARC method including its worst case evaluation complexity, global and asymptotic convergence. We describe our implementation and report numerical experience that confirms that our implementation of ARC _q K outperforms a classic Steihaug-Toint trust-region method, and the ARC method of the GALAHAD library. The latter solves the subproblem in nested Krylov subspaces by a Lanczos-based method, which requires the storage of a dense matrix that can be comparable to or larger than the two dense arrays required by our approach if the problem is large or requires many Lanczos iterations. Finally, we generalize our convergence results to inexact Hessians and nonlinear least-squares problems.