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The infinite-horizon optimal control problem of nonlinear discrete-time systems with actuator saturation is considered in this paper. In order to deal with actuator saturation, a novel nonquadratic functional is introduced, and the constrained generalized Hamilton-Jacobi-Bellman (GHJB) equation and Hamilton-Jacobi-Bellman (HJB) equation for nonlinear discrete-time systems are derived in terms of non-quadratic functionals. The optimal saturated controller is obtained by a novel iterative algorithm based on the constrained GHJB equation, and a convergence proof is presented, where a neural network is used to approximate the value function. Finally, a nearly optimal saturated controller is obtained. The effectiveness of this algorithm is demonstrated by a numerical example.