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We address an optimal stopping problem over the set of Bermudan-typestrategies $\Theta$ (which we understand in a more general sense than thestopping strategies for Bermudan options in finance) and with non-linearoperators (non-linear evaluations) assessing the rewards, under generalassumptions on the non-linear operators $\rho$. We provide a characterizationof the value family V in terms of what we call the ($\rho$,$\Theta$) -Snellenvelope of the pay-off family. We establish a Dynamic Programming Principle.We provide an optimality criterion in terms of a ($\rho$,$\Theta$) -martingaleproperty of V on a stochastic interval. We investigate the($\rho$,$\Theta$)-martingale structure and we show that the ''first time'' whenthe value family coincides with the pay-off family is optimal. The reasoningsimplifies in the case where there is a finite number n of pre-describedstopping times, where n does not depend on the scenario $\omega$. We provideexamples of non-linear operators entering our framework.

Optimal stopping: Bermudan strategies meet non-linear evaluations