Parallel Domain-Decomposition Algorithms for Complexity Certification of Branch-and-Bound Algorithms for Mixed-Integer Linear and Quadratic Programming

When implementing model predictive control (MPC) for hybrid systems with a linear or a quadratic performance measure, a mixed-integer linear program (MILP) or a mixed-integer quadratic program (MIQP) needs to be solved, respectively, at each sampling instant. Recent work has introduced the possibility to certify the computational complexity of branch-and-bound (B&B) algorithms when solving MILP and MIQP problems formulated as multi-parametric MILPs (mp-MILPs) and mp-MIQPs. Such a framework allows for computing the worst-case computational complexity of standard B&B-based MILP and MIQP solvers, quantified by metrics such as the total number of LP/QP iterations and B&B nodes. These results are highly relevant for real-time hybrid MPC applications. In this paper, we extend this framework by developing parallel, domain-decomposition versions of the previously proposed algorithm, allowing it to scale to larger problem sizes and enable the use of high-performance computing (HPC) resources. Furthermore, to reduce peak memory consumption, we introduce two novel modifications to the existing (serial) complexity certification framework, integrating them into the proposed parallel algorithms. Numerical experiments show that the parallel algorithms significantly reduce computation time while maintaining the correctness of the original framework.