In large directed complex networks, it may result unfeasible to successfully pinning control the whole network. Indeed, when the pinner node can be connected only to a limited number of nodes, it may be impossible to guarantee pinning controllability of all the network nodes. In this paper, we introduce the partial pinning control problem, which consists in determining the optimal selection of the nodes to be pinned so as to maximize the fraction of nodes of the whole network that can be asymptotically controlled to the pinner's trajectory. A suboptimal solution to this problem is provided for a class of nonlinear node dynamics, together with the bounds on the minimum coupling and control gains required to "partially control" the network. The theoretical analysis is translated into an integer linear program (ILP), which is solved on a testbed network of 688 nodes. (C) 2017 Elsevier Ltd. All rights reserved.