In the field of engineering vibration control, the parameter design of traditional dynamic vibration absorbers typically neglects the damping inherent in the primary system. However, structural damping is unavoidable in practical applications, and disregarding this factor introduces significant errors and diminishes vibration suppression effectiveness. To resolve this limitation and enhance engineering applicability, this study aims to solve the optimization design problem of a negative-stiffness dynamic vibration absorber incorporating an amplification mechanism under the condition of primary system damping. The research first establishes the precise governing differential equations of the system and derives its analytical solution. Given that the presence of primary system damping invalidates the classical fixed-point theory, a numerical optimization approach is employed: the primary system amplitude is normalized and based on the criterion of minimizing the maximum primary system amplitude, optimal parameters including the stiffness ratio and damping ratio are determined through numerical search techniques. The accuracy of the analytical solution is subsequently verified using numerical simulations. The results demonstrate that, compared to traditional dynamic vibration absorber designs ignoring primary system damping, the proposed method significantly improves the overall vibration reduction efficiency of the negative-stiffness dynamic vibration absorber with amplification mechanism and effectively reduces the sensitivity of the primary system's resonant amplitude to variations in excitation frequency. Comparative vibration suppression experiments between the grounded negative-stiffness dynamic vibration absorber with amplification mechanism and conventional dynamic vibration absorbers further validate that the proposed negative-stiffness device exhibits significantly superior performance in both effective bandwidth and vibration reduction depth. This study provides a solid theoretical foundation and a practical optimization methodology for negative-stiffness dynamic vibration absorbers incorporating amplification mechanisms. Its optimization strategy, which explicitly considers primary damping, markedly enhances the practical effectiveness and adaptability of the absorber. Consequently, the proposed negative-stiffness dynamic vibration absorber demonstrates broad application prospects in engineering fields requiring efficient broadband vibration suppression, such as precision instruments, offering a novel solution for high-performance vibration control.