Improved second-order evaluation complexity for unconstrained nonlinear optimization using high-order regularized models
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The unconstrained minimization of a sufficiently smooth objective function f(x) is considered, for which derivatives up to order p, p≥ 2, are assumed to be available. An adaptive regularization algorithm is proposed that uses Taylor models of the objective of order p and that is guaranteed to find a first- and second-order critical point in at most O (( ϵ_1^-p+1/p, ϵ_2^-p+1/p-1) ) function and derivatives evaluations, where ϵ_1 and ϵ_2 >0 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 p-th derivative tensor is Lipschitz continuous and that f(x) is bounded from below. The evaluation-complexity bound for second-order criticality improves on all such known existing results.
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