Convex optimization with inexact second-order oracles

In this thesis, we introduce a general inexact oracle and focus our study on the inexact second order and its influence on the Cubic Regularization of the Newton’s method. We show that, for an inexact second-order oracle of degree zero, the upper bound of the value of the objective function converges asymptotically, under certain conditions, to a non-accumulative error depending on the accuracy of the oracle. We compare it to existing results for the equivalent inexact first-order oracle and show that it could present, depending on the parameters, better results. We study the inexact second-order oracle of degree one applied to the same method and prove an upper bound of the objective function. We compare it to the one of the inexact second-order oracle of degree zero, but also to the newly found result of the inexact first-order oracle of degree one and its relaxation.