The combination of negative stiffness devices and inerters to traditional base isolators (TBI) and tuned mass dampers (TMD) does not exist in any state-of-the-art. Therefore, to pursue the research using the above-mentioned research scope, the negative stiffness inerter passive dampers such as negative stiffness inerter-based base isolators (NSIBI), negative stiffness base isolators (NSBI), negative stiffness inerter-based tuned mass dampers (NSITMD), and negative stiffness tuned mass dampers (NSTMD) are introduced in this paper. H2 and H∞ optimization methods are applied to derive the exact closed-form expressions for the optimal design parameters of these novel passive vibration dampers. Newton’s second law applies to derive the governing equations of motion of the controlled structures. The transfer function formation and Newmark-beta method are applied to determine the dynamic responses of the controlled structures analytically and numerically. Hence, H2 optimized NSIBI and NSBI have 45.98% and 46.71% more dynamic response reduction capacities than optimum TBI. In addition, H∞ optimized NSIBI and NSBI have 58.36% and 57.32% more dynamic response reduction capacities than optimum TBI. Furthermore, the optimum NSITMD and NSTMD have 0.42%, 10.84%, and 4.5%, 13.48% more dynamic response reduction capacities than traditional TMD. All the derivations are mathematically accurate. Display omitted •Combination of NSD and inerters to TBI and TMD does not exist in state-of-the-art.•Negative stiffness inerter passive dampers are introduced.•H2 and H∞ optimized design parameters are introduced.•Vibration reduction capacities of NSIBI and NSBI are 45% more than TBI.•NSITMD and NSTMD have 10% more vibration reduction capacities than TMD.