# Hölder Gradient Descent and Adaptive Regularization Methods in Banach Spaces for First-Order Points

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## Résumé

This paper considers optimization of smooth nonconvex functionals in smooth
infinite dimensional spaces. A Hölder gradient descent algorithm is first
proposed for finding approximate first-order points of regularized polynomial
functionals. This method is then applied to analyze the evaluation complexity of
an adaptive regularization method which searches for approximate first-order
points of functionals with beta-Hölder continuous derivatives. It is shown
that finding an epsilon-approximate first-order point requires at most
O(epsilon^{-(p+beta)/(p+\beta-1)})
evaluations of the functional and its first p derivatives.
langue originale Anglais Publié - 7 avr. 2021