• optimal control;
  • direct multiple shooting;
  • mixed-integer nonlinear optimization;
  • mixed-integer optimal control;
  • double-lane change manoeuvre;
  • gear choice


We present a numerical method and results for a recently published benchmark problem (Optim. Contr. Appl. Met. 2005; 26:1–18; Optim. Contr. Appl. Met. 2006; 27(3):169–182) in mixed-integer optimal control. The problem has its origin in automobile test-driving and involves discrete controls for the choice of gears. Our approach is based on a convexification and relaxation of the integer controls constraint. Using the direct multiple shooting method we solve the reformulated benchmark problem for two cases: (a) As proposed in (Optim. Contr. Appl. Met. 2005; 26:1–18), for a fixed, equidistant control discretization grid and (b) As formulated in (Optim. Contr. Appl. Met. 2006; 27(3):169–182), taking into account free switching times. For the first case, we reproduce the results obtained in (Optim. Contr. Appl. Met. 2005; 26:1–18) with a speed-up of several orders of magnitude compared with the Branch&Bound approach applied there (taking into account precision and the different computing environments). For the second case we optimize the switching times and propose to use an initialization based on the solution of (a). Compared with (Optim. Contr. Appl. Met. 2006; 27(3):169–182) we were able to reduce the overall computing time considerably, applying our algorithm. We give theoretical evidence on why our convex reformulation is highly beneficial in the case of time-optimal mixed-integer control problems as the chosen benchmark problem basically is (neglecting a small regularization term). Copyright © 2009 John Wiley & Sons, Ltd.