In order to assess the lifetime characteristics of highly reliable products, the step-stress accelerated degradation test (ADT) is a practical and effective solution, especially when there are very few items available for testing. During the past decades, the step-stress ADT has been studied by many researchers based on the assumption that the underlying degradation path follows one of the well-known but restricted stochastic processes such as Wiener, gamma, and inverse Gaussian. In practice, however, the degradation path of a product/device may not follow these specific processes, and the researchers are calling for a more flexible but unified approach toward generalized degradation models. To address this issue, the exponential dispersion process has been proposed, which is a generalized stochastic process including Wiener, gamma, and inverse Gaussian processes as special cases. In this work, we develop the step-stress ADT of products/devices when the underlying degradation path follows a class of the exponential dispersion processes. Based on this framework, the design optimization for the step-stress ADT is formulated under the C-optimality. Under the constraint that the total experimental cost does not exceed a pre-specified budget, the optimal design parameters such as measurement frequency and test termination time are determined via minimizing the approximate variance of the estimated mean time to failure of a product/device under the normal operating condition.