A mixed $H_{2}/H_{\infty}$ control problem under a sparsity constraint is investigated for multi-agent control systems (MAS) to provide robustness against model uncertainty and to reduce the communication cost. First, proximal alternating linearized minimization (PALM) is employed to develop a centralized social optimization algorithm, which is guaranteed to converge to a globally optimal sparse controller. Next, we investigate a noncooperative game that accommodates different control performance criteria of the agents and propose a best-response dynamics algorithm based on PALM. A special case of this game produces a partially distributed social optimization solution. We validate the proposed algorithms using a network with open-loop-unstable nodes and demonstrate superiority of the PALM-based method to a previously investigated sparsity-constrained mixed $H_{2}/H_{\infty}$ controller.