This paper presents a method to certify the computational complexity of a standard Branch and Bound method for solving Mixed-Integer Quadratic Programming (MIQP) problems defined as instances of a multi-parametric MIQP. Beyond previous work, not only the size of the binary search tree is considered, but also the exact complexity of solving the relaxations in the nodes by using recent results from exact complexity certification of active-set QP methods. With the algorithm proposed in this paper, a total worst-case number of QP iterations to be performed in order to solve the MIQP problem can be determined as a function of the parameter in the problem. An important application of the proposed method is Model Predictive Control for hybrid systems, that can be formulated as an MIQP that has to be solved in real-time. The usefulness of the proposed method is successfully illustrated in numerical examples.

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 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 performance measure for hybrid systems requires the solution of a mixed-integer linear program (MILP) at each time instance. A well-known method to solve MILP problems is branch-and-bound (B&B). To enhance the performance of B&B, start heuristic methods are often used, where they have shown to be useful supplementary tools to find good feasible solutions early in the B&B search tree, hence, reducing the overall effort in B&B to find optimal solutions. In this work, we extend the recently-presented complexity certification framework for B&B-based MILP solvers to also certify computational complexity of the start heuristics that are integrated into B&B. Therefore, the exact worst-case computational complexity of the three considered start heuristics and, consequently, the B&B method when applying each one can be determined offline, which is of significant importance for real-time applications of hybrid MPC. The proposed algorithms are validated by comparing against the corresponding online heuristic-based MILP solvers in numerical experiments.