In previous works we proposed the Reverse Zeldovich Approximation (RZA) method, which can be used to estimate the cosmological initial conditions underlying the galaxy distribution in the Local Universe using peculiar velocity data. In this paper, we apply the technique to run constrained cosmological simulations from the RZA-reconstructed initial conditions, designed to reproduce the large-scale structure of the Local Universe. We test the method with mock peculiar velocity catalogues extracted from a reference simulation. We first reconstruct the initial conditions of this reference simulation using the mock data, and then run the reconstructed initial conditions forward in time until z=0. We compare the resulting constrained simulations with the original simulation at z=0 to test the accuracy of this method. We also compare them with constrained simulations run from the mock data without the addition of RZA, i.e. using only the currently established constrained realizations (CR) method. Our re-simulations are able to correctly recover the evolution of the large-scale structure underlying the data. The results show that the addition of RZA to the CR method significantly improves both the reconstruction of the initial conditions and the accuracy of the obtained constrained resimulations. Haloes from the original simulation are recovered in the re-simulations with an average accuracy of about 2 Mpc/h on their position and a factor of 2 in mass, down to haloes with a mass of approx 10^14 M_odot/h. In comparison, without RZA the re-simulations recover only the most massive haloes with masses of about 5x10^14 M_odot/h and higher, and with a systematic shift on their position of about approx 10 Mpc/h due to the cosmic displacement field. We show that with the additional Lagrangian reconstruction step introduced by the RZA, this shift can be removed.