This paper focuses on the problem of distributed optimal coordination/consensus over continuous-time multi-agent systems, in which each agent communicates with other agents to cooperatively find the global minimum solution. The global objective is the sum of all local objective functions. To achieve this, based on the zero-gradient-sum technique and fixed-time control theory, an initialization-free distributed nonlinear consensus scheme is developed to achieve the fixed-time optimal consensus of all agents. In addition, the initial states are not desired to be the local optimal solution by using the sliding mode control technique. Furthermore, the determined upper bound of the convergence time is the zero-crossing point of a sine function, making it unrelated to the initial state information. This study provides three cases to illustrate the effectiveness of the developed scheme.