The stochastic optimization of a nonlinear energy sink (NES) with a time-dependent stiffness is considered. The NES is linearly coupled to a main system. The optimization aims to find the stiffness properties of the NES that minimize the expected value of the velocity of the main system while accounting for the statistical distributions of the excitation amplitude and frequency. It is shown that the system’s responses are highly sensitive to uncertainty and can even exhibit a discontinuous behavior. This represents a major hurdle for the optimization, which is already hampered by the potentially large computational cost associated with the time integrations. To tackle the high-sensitivity to uncertainties and reduce the computational burden, a dedicated surrogate-based stochastic optimization algorithm is used. Specifically, the approach uses Kriging surrogates built from the unsupervised identification of clusters resulting from response discontinuities. Comparisons between efficiencies of optimal nonlinear absorbers with and without time-dependent stiffness are performed and discussed.