This paper proposes a centralized secondary control for real-time steady-state optimization of multi-terminal high-voltage direct current (HVDC) grids, considering both voltage and current limits. This control begins with detailed dynamic models of key grid components, including modular multilevel converter (MMC) stations and their control layers, followed by the derivation of a quasi-static input-output model suitable for steady-state control. Using this model, a general optimization problem is formulated, and the associated Karush-Kuhn-Tucker (KKT) conditions are characterized. A secondary controller based on primal-dual dynamics is then proposed to adjust the voltage setpoints of dispatchable MMCs, ensuring convergence to a steady state that satisfies the optimal conditions. The inclusion of current constraints necessitates partial knowledge of the network model, which naturally supports a centralized framework. To reduce the communication burden, a communication triggering mechanism is introduced that limits message exchanges between the control center and MMC stations without degrading performance. The proposed controller is validated through case studies using an offshore multi-terminal HVDC grid with heterogeneous MMC stations, simulated in MATLAB/Simulink. Results confirm that the proposed controller drives the system to optimal operation, while significantly reducing the communication burden without compromising performance.