This paper presents a non-linear model for the optimal power flow in multi-terminal high-voltage direct current transmission systems. The proposed methodology considers exclusively the DC side of the grid and includes branches with DC/DC converters. Losses of lines and DC/DC converters are also considered in the optimization model. A semidefinite approximation is proposed in order to obtain an unique solution for real time operation. The methodology is evaluated on a reduced version of the CIGRE B4 DC grid test system. Results demonstrate the proposed approximation is efficient and accurate compared to the non-linear model solved in GAMS.