Improved worst-case evaluation complexity for potentially rank-deficient nonlinear least-Euclidean-norm problems using higher-order regularized models

Given a sufficiently smooth vector-valued function r(x), a local minimizer of ‖r(x)‖2 within a closed, non-empty, convex set F is sought by modelling ‖r(x)‖q2/q with a p-th order Taylorseries approximation plus a (p + 1)-st order regularization term for given even p and some appropriate associated q. The resulting algorithm is guaranteed to find a value x̄ for which ‖r(x̄)‖2 ≤ ǫp or χ(x̄) ≤ ǫd, for some first-order criticality measure χ(x) of ‖r(x)‖2 within F , using at most O(max{max(ǫd, χmin) ,max(ǫp, rmin) −1/2i}) evaluations of r(x) and its derivatives; here rmin and χmin ≥ 0 are any lower bounds on ‖r(x)‖2 and χ(x), respectively, and 2 is the highest power of 2 that divides p. An improved bound is possible under a suitable full-rank assumption.