In this work, we investigate on minimizing the energy consumption of a wireless backhaul communication network through a joint optimization problem 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 which 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 stepwise cost functions which is very hard to optimize. We then introduce a linear relaxation of the problem, which exploits the convexity of the energy cost as a function of the throughput on a radio link. This yields lower bounds on the energy consumption, and eventually a heuristic algorithm based on the fractional optimum is presented. Our models are validated through extensive experiments which are reported and discussed. The results of the simulations testify the potentialities behind this novel approach. In particular, our algorithm takes a good advantage of the convexity of the cost function, inducing a quite small integrity gap in practice.