Legged robots are typically in rigid contact with the environment at multiple locations, which add a degree of complexity to their control. We present a method to control the motion and a subset of the contact forces of a floating-base robot. We derive a new formulation of the lexicographic optimization problem typically arising in multi-task motion/force control frameworks. The structure of the constraints of the problem (i.e. the dynamics of the robot) allows us to find a sparse analytical solution. This leads to an equivalent optimization with reduced computational complexity, comparable to inverse-dynamics based approaches. At the same time, our method preserves the flexibility of optimization based control frameworks. Simulations were carried out to achieve different multi-contact behaviors on a 23-degree-of-freedom humanoid robot, validating the presented approach. A comparison with another state-of-the-art control technique with similar computational complexity shows the benefits of our controller, which can eliminate force/torque discontinuities.