Summary: In this paper, we study a distributed constrained optimal consensus problem for discrete-time first-order integrator systems under bounded input. Each agent is assigned with a local convex cost function, and all agents are required to achieve consensus at the minimum of the aggregate cost over a common convex constraint set, which is only accessible by part of the agents. A 2-step control protocol is designed to solve the problem under bounded input. Firstly, at each time step, each agent moves a bounded step of the subgradient descent from an individual cost and another one along the projection direction if it can access the constraint. The second movement is then given by a bounded average of the relative positions to neighbors. Specifically, to coordinate the subgradient step within the network without using global information, we introduce an estimate of the upper bound of all agents’ subgradients, which is updated by a unilateral consensus mechanism. Under the given control protocol, we obtain the necessary and sufficient conditions to achieve the constrained optimal consensus for a fixed topology. Under similar conditions, we also solve the problem for switching topologies and conduct a convergence rate analysis for strongly convex costs.