In this paper, we consider the global optimal consensus problem for discrete-time multi-agent systems with bounded control protocols over a fixed and directed communication network. Each agent is described by a discrete-time single integrator and endowed with a quadratic objective function which is private to itself. For each agent, we develop two bounded distributed protocols: bounded proportional-integral (PI) protocol and bounded integral (I) protocol, based on the information received from its neighboring agents through the communication network and the gradient of its own objective function. We show that the proposed hounded distributed protocols with properly chosen parameters solve the global optimal consensus problem, i.e., the multi-agent system achieves consensus at a state that minimizes the sum of the objective functions, if the directed communication network is strongly connected and detailed balanced. (C) 2018 Elsevier Ltd. All rights reserved,