In this work, we investigate on minimizing the energy consumption of a wireless backhaul communication network through a joint optimization of data routing and radio configuration. The backhaul network is modeled by a digraph in which the nodes represent radio base stations and the arcs denote radio links. According to the scenario under consideration, a power-efficient configuration can be characterized by a modulation constellation size and a transmission power level. Every link holds a set of power-efficient configurations, each of them associating a capacity with its energy cost. The optimization problem involves deciding the network's configuration and flows that minimize the total energy expenditure, while handling all the traffic requirements simultaneously. An exact mathematical formulation of the problem is presented. It relies on a minimum cost multicommodity flow with step increasing cost functions, which is very hard to optimize. We then propose a piecewise linear convex function, obtained by linear interpolation of power-efficient configuration points, that provides a good approximation of the energy consumption on the links, and present a relaxation of the previous formulation that exploits the convexity of the energy cost functions. This yields lower bounds on the energy consumption, and finally a heuristic algorithm based on the fractional optimum is employed to produce feasible solutions. Our models are validated through extensive experiments that are reported and discussed. The results verify the potentialities behind this novel approach. In particular, our algorithm induces a satisfactory integrality gap in practice.