A novel family of high-order structure-preserving methods is proposed for the nonlinear Schrödinger equation. The methods are developed by applying the multiple relaxation idea to the different Runge--Kutta methods. It is shown that the multiple relaxation  Runge--Kutta methods can achieve high-order accuracy in time and preserve multiple original invariants at the discrete level. Several numerical experiments are carried out to support the theoretical results and illustrate the effectiveness and efficiency of the proposed methods.