A regularization algorithm using inexact function values and inexact
derivatives is proposed and its evaluation complexity analyzed. This algorithm
is applicable to unconstrained problems and to problems with inexpensive
constraints (that is constraints whose evaluation and enforcement has
negligible cost) under the assumption that the derivative of highest degree is
β-H\"{o}lder continuous. It features a very flexible adaptive mechanism
for determining the inexactness which is allowed, at each iteration, when
computing objective function values and derivatives. The complexity analysis
covers arbitrary optimality order and arbitrary degree of available approximate
derivatives. It extends results of Cartis, Gould and Toint (2018) on the
evaluation complexity to the inexact case: if a qth order minimizer is sought
using approximations to the first p derivatives, it is proved that a suitable
approximate minimizer within ϵ is computed by the proposed algorithm
in at most O(ϵ−p−q+βp+β) iterations and at most
O(∣log(ϵ)∣ϵ−p−q+βp+β) approximate
evaluations. An algorithmic variant, although more rigid in practice, can be
proved to find such an approximate minimizer in
O(∣log(ϵ)∣+ϵ−p−q+βp+β) evaluations.While
the proposed framework remains so far conceptual for high degrees and orders,
it is shown to yield simple and computationally realistic inexact methods when
specialized to the unconstrained and bound-constrained first- and second-order
cases. The deterministic complexity results are finally extended to the
stochastic context, yielding adaptive sample-size rules for subsampling methods
typical of machine learning.Comment: 32 page