In large directed complex networks, it may result unfeasible to successfully pinning control the whole network. Indeed, when the number of pinned nodes has to be lower than the number of strictly connected components in the graph, it is impossible to guarantee pinning controllability of all the nodes in the network. In this paper, we introduce the partial pinning control problem, which consists in determining the optimal selection of the pinned nodes so as to maximize the fraction of nodes that can be asymptotically controlled to the pinner's trajectory. A suboptimal solution to this problem is provided for a class of nonlinear nodes' dynamics, together with the bounds on the minimum coupling and control gains required to actually control the network. The theoretical analysis is translated into an integer linear programming (ILP) problem, which is solved on a testbed network of 688 nodes building an ad hoc algorithm in Matlab.