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Mixed‐integer model predictive control (MI‐MPC) can be a powerful tool for controlling hybrid systems. In case of a linear‐quadratic objective in combination with linear or piecewise‐linear system dynamics and inequality constraints, MI‐MPC needs to solve a mixed‐integer quadratic program (MIQP) at each sampling time step. This paper presents a collection of exact block‐sparse presolve techniques to efficiently remove decision variables, and to remove or tighten inequality constraints, tailored to mixed‐integer optimal control problems. In addition, we describe a novel approach based on a heuristic presolve algorithm to compute a feasible but possibly suboptimal MIQP solution. We present benchmarking results for a C code implementation of the proposed  BB‐ASIPM  solver, including a branch‐and‐bound (B&amp;B) method with the proposed tailored presolve techniques and an active‐set based interior point method (ASIPM), compared against multiple state‐of‐the‐art MIQP solvers on a case study of motion planning with obstacle avoidance constraints. Finally, we demonstrate the feasibility and computational performance of the  BB‐ASIPM  solver in embedded system on a dSPACE Scalexio real‐time rapid prototyping unit for a second case study of stabilization for an underactuated cart‐pole with soft contacts.

Tailored presolve techniques in branch‐and‐bound method for fast mixed‐integer optimal control applications