Overall Complexity Certification of a Standard Branch and Bound Method for Mixed-Integer Quadratic Programming

TL;DR

This paper introduces a method to certify the worst-case computational complexity of a standard Branch and Bound algorithm for MIQP problems, considering both search tree size and relaxation solving complexity, with applications in real-time hybrid systems control.

Contribution

It extends previous complexity analysis by incorporating the exact complexity of solving relaxations using recent active-set QP certification results, enabling precise worst-case iteration bounds.

Findings

Provides a way to determine total worst-case QP iterations based on problem parameters

Demonstrates the method's effectiveness through numerical examples

Applicable to real-time Model Predictive Control for hybrid systems

Abstract

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 result 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.