In this paper, we present a method to exactly certify the computational complexity of standard suboptimal branch-and-bound (B&B) algorithms for computing suboptimal solutions to mixed-integer linear programming (MILP) problems. Three well-known approaches for suboptimal B&B are considered. This work shows that it is possible to exactly certify the computational complexity also when these approaches are used. Moreover, it also enables to compute exact bounds on the level of suboptimality actually to be obtained online, also for methods previously without any such guarantees. It additionally provides a novel deeper insight into how they affect the performance of the B&B algorithm in terms of the required computation time and memory storage. The exact bounds on the online worst-case computational complexity (e.g., the accumulated number of LP solver iterations or size of the B&B tree) and the worst-case suboptimality computed with the proposed method are very relevant for real-time applications such as Model Predictive Control (MPC) for hybrid systems. The numerical experiments confirm the correctness of the proposed method, and they demonstrate the usefulness of the certification method for certification of a standard online B&B-based MILP solver employing the three considered suboptimal techniques. Copyright (c) 2023 The Authors.

Model predictive control (MPC) with linear cost function for hybrid systems requires the solution of a mixed-integer linear program (MILP) at each sampling time. The branch and bound (B&B) method is a commonly used tool for solving mixed-integer problems. In this work, we present an algorithm to exactly certify the computational complexity of a standard B&B-based MILP solver. By the proposed method, guarantees on worst-case complexity bounds, e.g., the worst-case iterations or size of the B&B tree, are provided. This knowledge is a fundamental requirement for the implementation of MPC in a real-time system. Different node selection strategies, including best-first, are considered when certifying the complexity of the B&B method. Furthermore, the proposed certification algorithm is extended to consider warm-starting of the inner solver in the B&B. We illustrate the usefulness of the proposed algorithm by comparing against the corresponding online MILP solver in numerical experiments using both cold-started and warm-started LP solvers.

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.