This paper presents a multi-objective design optimization of axial slot casing treatments in a 1.5 stage transonic compressor. To perform the optimization, an in-house optimization design platform is constructed using genetic algorithm and Kriging surrogate model. The optimization objectives are the peak efficiency at the design rotating speed and the stall margin improvements at both design and off-design rotating speeds. Instead of massive unsteady simulations that are required to accurately predict the stall margin, a stall margin improvement indicator is introduced based on the time-averaged axial momentum budget analysis at the rotor tip region. With this indicator, only one steady simulation is needed to evaluate the stall margin enhancement ability of a new casing treatment design at a given rotating speed. The axial slots are parameterized with a total of eight design parameters. The meridional profiles are described with two B-spline curves to define parameters such as the size, axial position, and shape of the slots. Circumferential parameters include the open area ratio and inclination angle. When the optimization finally converges, different optimal variants are selected from the database and their effects are investigated with both experiments and numerical simulations. The detailed flow fields are analyzed in depth with results from unsteady simulations. The application of the optimal casing treatments exerts a suction and re-injection effect, pushes the interface between tip leakage flow and incoming main flow downstream and reduced flow blockage in the blade tip region. Consequently, the stability of the compressor is significantly improved.