Here we present our latest studies of relativistic jet formation in the vicinity of a rotating black hole where the reconnection process has been taken into account. In order to simplify the problem, we use Lagrangian formalism and develop a method which enables us to consider a magnetized plasma as a set of magnetic flux tubes [5, 6]. Within the limits of the Lagrangian approach, we perform numerical simulations of the flux tube (nonlinear string) behavior which clearly demonstrates the process of relativistic jet formation in the form of outgoing torsional nonlinear aves. It turns out that the jet is produced deep inside the ergosphere where the flux tube takes away spinning energy from the black hole due to the nonlocal Penrose process [2]. This is similar to the Blandford-Znajek (BZ) mechanism to some extent [8], however, the string mechanism is essentially time dependent. It is shown that the leading part of the accreting tube gains negative energy and therefore has to stay in the ergosphere forever. Simultaneously, another part of the tube propagates along the spinning axis away from the hole with nearly the speed of light. As a result, the tube is continuously stretching and our mechanism is essentially time dependent. Obviously, such process cannot last infinitely long and we have to take into account the reconnection process. Due to reconnection, the topology of the flux tube is changed and it gives rise to a plasmoid creation which propagates along spin axis of the hole with relativistic speed carrying off the energy and angular momentum away from the black hole.