Compared to the traditional constant friction TMD, the variable friction TMD exhibits linearity and stable mitigation capacity in both the time domain and frequency domain due to the core functionality of hysteretic damping tuned mass damper (HD-TMD). However, the designs of HD-TMD exhibit a wide range of diversity and incompleteness. To resolve this problem, this paper proposes optimal solutions for the parameters of HD-TMD in primary systems subjected to harmonic and white noise random excitations with force and based acceleration inputs. The availability of the closed-form solutions for complex systems with high accuracy in engineering is investigated, where the average errors for the optimal frequency ratio and optimal HD ratio are 0.65% and 3.22%, respectively, showcasing the effectiveness of closed-form solutions within the field of civil engineering. The impact of inherent structural damping is thoroughly explored. Accordingly, numerical results illustrate the seismic performance of the optimal HD-TMD in terms of both maximum response and standard deviation response, materializing the significance in practical engineering.