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A significant challenge in the modeling of short pulse fiber lasers is that with each successive generation there has been a dramatic increase in the amount by which the pulse varies over each round trip. Therefore, lumped rather than averaged models are required to accurately compute the periodically stationary (breather) solutions generated by these lasers. We use a spectral method to assess the linear stability of periodically stationary pulses in lumped models. This approach extends previous work by Menyuk and Wang on stationary pulses in averaged models. We first present a gradient based optimization method inspired by the work of Ambrose and Wilkening to compute periodically stationary pulses. Then, we use Floquet theory to characterize the linear stability of the pulses obtained using optimization in terms of the spectrum of the monodromy operator,M, obtained by linearization of the round trip operator about a periodically stationary pulse.