<p>An approximate solution for the constrained variational problem describing the start-up mode of the tower crane slewing mechanism during steady load hoisting has been developed. The root mean square value of the slewing drive torque has been selected as optimization criterion. The drive torque, its rate of change, and the drive power have been constrained. A dynamic model describing the coupled motion of the hoisting and slewing mechanisms has been formulated in the form of a system of ordinary differential equations. The approximate solution to the optimization problem has been represented by two polynomial functions: the first one ensures the satisfaction of the prescribed boundary conditions of motion, while the second one minimizes the optimization criterion. A modified particle swarm optimization algorithm has been employed to determine the optimal solution. As a result, pendulum-type load oscillations as well as oscillations in the structural elements of the crane have been significantly reduced.</p>