On Complexity Certification of Branch-and-Bound Methods for MILP and MIQP with Applications to Hybrid MPC

Abstract: In model predictive control (MPC), an optimization problem is solved at each time step, in which the system dynamics and constraints can directly be taken into account. The MPC concept can be further extended to the control of hybrid systems, where a part of the state and control variables has a discrete set of values. When applying MPC to linear hybrid systems with performance measures based on the 1-norm or the∞-norm, the resulting optimal control problem can be formulated as a mixed-integer linear program (MILP), while the optimal control problem with a quadratic performance measure can be cast as a mixed-integer quadratic program (MIQP). An efficient method to solve these non-convex MILP and MIQP problems is branch and bound (B&B) which relies on solving convex relaxations of the problem ordered in a binary search tree. For the safe and reliable real-time operation of hybrid MPC, it is desirable to have a priori guarantees on the worst-case complexity such that the computational requirements of the problem do not exceed the time and hardware capabilities.Motivated by this need, this thesis aims to certify the computational complexity of standard B&B methods for solving MILPs and MIQPs in terms of, e.g., the size of the search tree or the number of linear systems of equations (iterations) that are needed to be solved online to compute optimal solution. In particular, this knowledge enables us to compute relevant worst-case complexity bounds for the B&B-based MILP and MIQP solvers, which has significant importance in, e.g., real-time hybrid MPC where hard real-time requirements have to be fulfilled. The applicability of the proposed certification method is further extended to suboptimal B&B methods for solving MILPs, where the computational effort is reduced by relaxing the requirement to find a globally optimal solution to instead finding a suboptimal solution, considering three different suboptimal strategies. Finally, the proposed framework is extended to the cases where the performance of B&B is enhanced by considering three common start heuristic methods that can help to find good feasible solutions early in the B&B search process.