The nonlinear negative stiffness inertial amplifier base isolators (NNSIABI) for vibration reduction of dynamic systems 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 novel isolators. For SDOF system, the dynamic response reduction capacity of H2 and H∞ optimized linear NSIABI are significantly 64.78% and 77.14% superior to the optimum traditional base isolators. In contrast, the dynamic response reduction capacities of H2 and H∞-optimized nonlinear NNSIABI are significantly 64.66% and 66.56% superior to the optimum traditional base isolators. For MDOF systems, the optimum linear NSIABI’s dynamic response reduction capacities are significantly 95.06% and 97.80%, superior to the optimum traditional base isolators. In addition, the optimum nonlinear NNSIABI are significantly 94.88% and 97.68%, superior to the traditional base isolators. All the results are mathematically corrected and applicable for practical applications.