Abstract: To achieve simultaneous control of inter-storey drift and floor acceleration, a self-centering tuned viscoelastic mass damper (STVMD) is proposed. Using shape memory alloy bars to provide self-centering and energy dissipation capabilities, the damper is integrated with an inertial system and incorporates viscoelastic energy-dissipation materials to provide controllable damping. Based on the mechanical model of the damper using an equivalent linearization method, the mean-square responses of displacements and of accelerations for the structure-STVMD system under random excitations are derived. To obtain reasonable design parameters, a parameter sensitivity study and a multi-objective optimization are conducted using the displacement and acceleration vibration-reduction ratios and the damping-deformation enhancement ratio as evaluation indices. Based on the optimized parameters, a numerical model for the controlled structure is developed, and nonlinear time-history analyses are performed to verify the seismic mitigation effectiveness of the damper. Research results indicate that increasing the inertance-to-mass ratio and the supplemental damping ratio can further reduce displacement and acceleration responses, but may weaken the damping enhancement effect. The STVMD significantly mitigates seismic responses. Under rare earthquakes, the peak displacement reduction reaches 30.7% and the maximum acceleration reduction reaches 34.5%. Moreover, the deformation of energy-dissipation components inside the damper is amplified by approximately 2.6 times relative to the structural displacement. The hysteresis curves are stable and full, demonstrating favorable self-centering and energy dissipation capacities.