This paper investigates convex optimization strategies for coordinating a large-scale team of fully actuated mobile robots. Our primary motivation is both algorithm scalability as well as real-time performance. To accomplish this, we employ a formal definition from shape analysis for formation representation and repose the motion planning problem to one of changing (or maintaining) the shape of the formation. We then show that optimal solutions, minimizing either the total distance or minimax distance the nodes must travel, can be achieved through second-order cone programming techniques. We further prove a theoretical complexity for the shape problem of O(m1.5) as well as O(m) complexity in practice, where m denotes the number of robots in the shape configuration. Solutions for large-scale teams (1000's of robots) can be calculated in real time on a standard desktop PC. Extensions integrating both workspace and vehicle motion constraints are also presented with similar complexity bounds. We expect these results can be generalized for additional motion planning tasks, and will prove useful for improving the performance and extending the mission lives of large-scale robot formations as well as mobile ad hoc networks.