Complex networks can be considered as abstractions of complex systems existing in the real world. The potential functionality of networks is related to mesoscale structures in networks, among which the representative ones are community structure and core-periphery (CP) structure. Since many real-world networks will inevitably be attacked, it is of great significance to enhance robustness of networks. However, few of the existing studies about robustness have laid emphasis on robustness of CP structure. In this brief, we first propose a new index to measure the ability of CP structure to resist attacks or errors. Several efficient algorithms based on this index are then devised to maximize robustness of CP structure under reasonable constraint. Numerical results show that the robustness of the CP structure of several representative real-world networks is markedly enhanced. The structural changes in the optimized networks under study and their implication are also discussed.