Optimal path planning is prone to convergence to local, rather than global, optima. This is often the case for mobile manipulators due to nonconvexities induced by obstacles, robot kinematics and constraints. This paper focuses on planning under end effector path constraints and attempts to circumvent the issue of converging to a local optimum. We propose a pipeline that first discovers multiple homotopically distinct paths, and then optimizes them to obtain multiple distinct local optima. The best out of these distinct local optima is likely to be close to the global optimum. We demonstrate the effectiveness of our pipeline in the optimal path planning of mobile manipulators in the presence of path and obstacle constraints.