The design and construction of network structures oriented towards different applications has attracted much attention recently. The existing studies indicated that structural heterogeneity plays different roles in promoting cooperation and robustness. Compared with rewiring a predefined network, it is more flexible and practical to construct new networks that satisfy the desired properties. Therefore, in this paper, we study a method for constructing robust cooperative networks where the only constraint is that the number of nodes and links is predefined. We model this network construction problem as a multi-objective optimization problem and propose a multi-objective evolutionary algorithm, named MOEA-Netrc, to generate the desired networks from arbitrary initializations. The performance of MOEA-Netrc is validated on several synthetic and real-world networks. The results show that MOEA-Netrc can construct balanced candidates and is insensitive to the initializations. MOEA-Netrc can find the Pareto fronts for networks with different levels of cooperation and robustness. In addition, further investigation of the robustness of the constructed networks revealed the impact on other aspects of robustness during the construction process.