In two recent papers regularization methods based on Taylor polynomial models for minimization were proposed that only rely on H\"older conditions on the higher order employed derivatives. Grapiglia and Nesterov considered cubic regularization with a sufficient descent condition that uses the current gradient and resembles the classical Armijo's criterion. Cartis, Gould, and Toint used Taylor models with arbitrary-order regularization and defined methods that tackle convex constraints employing the descent criterion that compares actual reduction with predicted reduction. The methods presented in this paper consider general (not necessarily Taylor) models of arbitrary order, employ a very mild descent criterion and handles general, non necessarily convex, constraints. Complexity results are compatible with the ones presented in the papers mentioned above.