Many problems in robotics require reasoning jointly over continuous dynamics and discrete events, such as making and breaking contact in manipulation and locomotion. Locally, these problems are well modeled by quadratic programs with complementarity constraints (QPCCs), a highly expressive class that can encode contact, friction, and more generally discrete or switching events. QPCCs are, however, non-convex, and complementarity constraints violate the constraint qualifications that standard nonlinear solvers rely on, and few methods compute quality, local solutions fast enough for use in planning and control pipelines. We observe that the relaxed complementarity constraints form a Lie group, and exploit this structure to perform optimization directly on the relaxed complementarity constraint manifold. Building on this idea, we introduce a numerically well-behaved retraction map that satisfies the relaxed complementarity constraints by construction, avoiding the ill-conditioning and degeneracy that impede classical formulations. We implement this approach in Marble, an open-source QPCC solver written in C++ with Julia and Python bindings, and show that it is faster and more robust than state-of-the-art baselines across a standard benchmark suite and a range of robotics problems, including several on which existing solvers fail to converge.