This note studies the self-triggered MPC problem for constrained discrete-time nonlinear systems. A novel self-triggered MPC algorithm enabling the codesign of triggering and control is proposed, in which the communication cost is explicitly considered in the total cost function, and both the control and triggering intervals can be optimally designed. For constrained nonlinear systems, the theoretical conditions on ensuring feasibility and closed-loop stability are developed, and the achieved control and communication performance is proved to be guaranteed. In particular, the parameter design approaches for satisfying feasibility and stability conditions are further presented in the case of quadratic cost functions. Furthermore, stronger results are provided for linear case. Finally, the effectiveness and advantages are verified via simulation and comparison studies.