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In a Hilbert space setting, for convex optimization, we show the convergenceof the iterates to optimal solutions for a class of accelerated first-orderalgorithms. They can be interpreted as discrete temporal versions of aninertial dynamic involving both viscous damping and Hessian-driven damping. Theasymptotically vanishing viscous damping is linked to the accelerated gradientmethod of Nesterov while the Hessian driven damping makes it possible tosignificantly attenuate the oscillations. By treating the Hessian-drivendamping as the time derivative of the gradient term, this gives, in discretizedform, first-order algorithms. These results complement the previous work of theauthors where it was shown the fast convergence of the values, and the fastconvergence towards zero of the gradients.

Convergence of iterates for first-order optimization algorithms with inertia and Hessian driven damping