An adaptive regularization method in Banach spaces

This paper considers optimization of nonconvex functionals in smooth infinite dimensional spaces. It is first proved that functionals in a class containing multivariate polynomials augmented with a sufficiently smooth regularization can be minimized by a simple linesearch-based algorithm. Sufficient smoothness depends on gradients satisfying a novel two-terms generalized Lipschitz condition. A first-order adaptive regularization method applicable to functionals with beta-Holder continuous derivatives is then proposed, that uses the linesearch approach to compute a suitable trial step. It is shown to find an is an element of -approximate first-order point in at most O( is an element of - p+ ss/p+ss-1) evaluations of the functional and its first p derivatives.