A key element of hydrologic routing models is that the discharge-storage relationship is assumed to follow a certain mathematical form, usually a linear or a power function, with parameters calibrated based on existing inflow-outflow data. This assumption simplifies the model calibration process but also constrains model operation throughout the flow range, potentially introducing biases. We present a new nonlinear hydrologic river routing approach where functions are only required to be nondecreasing. River reaches are modeled as conceptual reservoir cascades, with discharge-storage and loss/gain functions identified by the data. A novel parameter estimation approach is developed to identify these functions and other model parameters within a dynamical optimization framework. It is shown that hydrologic routing functions indeed exhibit different mathematical forms at different regions of their active range, and that the new approach is reliable, efficient, and robust under observational uncertainty. The model is demonstrated in lake and river routing applications for the Nile River, and it is also applicable for the estimation of nonlinear, nondecreasing functional relationships of general dynamic systems in state-space form.