Mathematical Programs with Blocks of Vanishing Constraints Arising in Discretized Mixed-Integer Optimal Control Problems

A time transformation technique is used to discretize an optimal control problem with discrete-valued controls and mixed control-state constraints, which depend on the discrete-valued controls. The discretized optimal control problem leads to a mathematical program with blocks of vanishing constraints (MPBVC). The MPBVC is analyzed in terms of stationarity concepts and constraint qualifications extending previous results on mathematical programs with vanishing constraints to the case of blocks of vanishing constraints. A relaxation technique is employed to regularize the MPBVC and to solve it numerically. The relaxation method is validated for a flight path optimization problem taking into account a discrete number of positions for the flaps of the aircraft in a landing approach.