This work considers the problem of optimal lane changing in a structured multi-agent road environment. A novel motion planning algorithm that can capture long-horizon dependencies as well as short-horizon dynamics is presented. Pivotal to our approach is a geometric approximation of the long-horizon combinatorial transition problem which we formulate in the continuous time-space domain. Moreover, a discrete-time formulation of a short-horizon optimal motion planning problem is formulated and combined with the long-horizon planner. Both individual problems, as well as their combination, are formulated as MIQP and solved in real-time by using state-of-the-art solvers. We show how the presented algorithm outperforms two other state-of-the-art motion planning algorithms in closed-loop performance and computation time in lane changing problems. Evaluations are performed using the traffic simulator SUMO, a custom low-level tracking model predictive controller, and high-fidelity vehicle models and scenarios, provided by the CommonRoad environment.