We are therefore concerned in this talk by optimisation algorithms that can exploit computations of function and  gradient values with controlled accuracy, that also enjoy satisfactory convergence  properties and complexity bounds.

We will first consider the case of convex quadratic optimization problems where we show how to exploit inaccurate matrix-vector products while maintaining a given accuracy threshold for the solution. Numerical experiments suggest that the new methods have significant potential, including in the increasingly more important context of multi-precision computations. We will next consider general nonconvex, then nonconvex nonsmooth optimization problems. An adaptive regularization algorithm using inexact functions and derivatives evaluations is proposed that needs $O(|\log(\epsilon)|\,\epsilon^{-2})$ evaluations of the problem's functions and their derivatives for finding an $\epsilon$-approximate first-order stationary point composite optimization problem.