2 research outputs found
Improved second-order evaluation complexity for unconstrained nonlinear optimization using high-order regularized models
The unconstrained minimization of a sufficiently smooth objective function
is considered, for which derivatives up to order , , are
assumed to be available. An adaptive regularization algorithm is proposed that
uses Taylor models of the objective of order and that is guaranteed to find
a first- and second-order critical point in at most
function and derivatives evaluations, where and
are prescribed first- and second-order optimality tolerances. Our approach
extends the method in Birgin et al. (2016) to finding second-order critical
points, and establishes the novel complexity bound for second-order criticality
under identical problem assumptions as for first-order, namely, that the -th
derivative tensor is Lipschitz continuous and that is bounded from
below. The evaluation-complexity bound for second-order criticality improves on
all such known existing results