Recently, many researchers have adopted the cycle-based degradation model to estimate the aging of battery energy storage system (BESS) based on the cycle information obtained from the rainflow counting method. However, the lack of a mathematical expression for the rainflow counting method is one of the biggest obstacles to the widespread use of the cycle-based degradation model. This paper deduces the exact analytical subgradients of the cycle-based aging cost function, which makes it possible to efficiently solve the optimal operation problem without the mathematical expression of the rainflow counting method. We introduce the subgradient projection algorithm to get the theoretical optimal operation in some special cases where the charge-discharge complementarity constraints can be relaxed. We also develop a heuristic algorithm based on the subgradient method to avoid simultaneous charging and discharging in general cases. The effectiveness of the proposed methodologies has been demonstrated with extensive numerical experiments. In the scenarios discussed in this paper, implementing the cycle-based operation rather than the energy-based operation reduces the aging cost by about 12%. When the duration of the simulation is 60 minutes, the computation time of a general-purpose solver is about 74 minutes, and the proposed algorithm only needs 21 seconds.