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A Mixed Integer Dual Quadratic Programming Algorithm Tailored for MPC
Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.ORCID iD: 0000-0001-6957-2603
Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
2006 (English)In: Proceedings of the 45th IEEE Conference on Decision and Control, 2006, 5693-5698 p.Conference paper (Refereed)
Abstract [en]

The objective of this work is to derive an MIQP solver tailored for MPC. The MIQP solver is built on the branch and bound method, where QP relaxations of the original problem are solved in the nodes of a binary search tree. The difference between the subproblems is often small and therefore it is interesting to be able to use a previous solution as a starting point in a new subproblem. This is referred to as a warm start of the solver. Because of its good warm start properties, a dual active set QP method was chosen. The method is tailored for MPC by solving a part of the KKT system using a Riccati recursion, which makes the computational complexity of the QP iterations grow linearly with the prediction horizon. Simulation results are presented both for the QP solver itself and when it is incorporated as a part of the MIQP solver. In both cases the computational complexity is significantly reduced compared to if a primal active set solver not utilizing structure is used.

Place, publisher, year, edition, pages
2006. 5693-5698 p.
Keyword [en]
Riccati equations, Computational complexity, Integer programming, Iterative methods, Predictive control, Quadratic programming, Tree searching, KKT system, QP iterations, QP relaxations, Riccati recursion, Binary search tree, Branch and bound method, Computational complexity, Mixed integer dual quadratic programming algorithm, Model predictive control, Prediction horizon
National Category
Control Engineering
URN: urn:nbn:se:liu:diva-12904DOI: 10.1109/CDC.2006.377215ISBN: 1-4244-0171-2OAI: diva2:17354
45th IEEE Conference on Decision and Control, San Diego, CA, USA, December, 2006
Available from: 2008-03-18 Created: 2008-03-18 Last updated: 2016-08-31
In thesis
1. Integer Quadratic Programming for Control and Communication
Open this publication in new window or tab >>Integer Quadratic Programming for Control and Communication
2008 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The main topic of this thesis is integer quadratic programming with applications to problems arising in the areas of automatic control and communication. One of the most widespread modern control methods is Model Predictive Control (MPC). In each sampling time, MPC requires the solution of a Quadratic Programming (QP) problem. To be able to use MPC for large systems, and at high sampling rates, optimization routines tailored for MPC are used. In recent years, the range of application of MPC has been extended to so-called hybrid systems. Hybrid systems are systems where continuous dynamics interact with logic. When this extension is made, binary variables are introduced in the problem. As a consequence, the QP problem has to be replaced by a far more challenging Mixed Integer Quadratic Programming (MIQP) problem, which is known to have a computational complexity which grows exponentially in the number of binary optimization variables. In modern communication systems, multiple users share a so-called multi-access channel. To estimate the information originally sent, a maximum likelihood problem involving binary variables can be solved. The process of simultaneously estimating the information sent by multiple users is called Multiuser Detection (MUD). In this thesis, the problem to efficiently solve MIQP problems originating from MPC and MUD is addressed. Four different algorithms are presented. First, a polynomial complexity preprocessing algorithm for binary quadratic programming problems is presented. By using the algorithm, some, or all, binary variables can be computed efficiently already in the preprocessing phase. In numerical experiments, the algorithm is applied to unconstrained MPC problems with a mixture of real valued and binary valued control signals, and the result shows that the performance gain can be significant compared to solving the problem using branch and bound. The preprocessing algorithm has also been applied to the MUD problem, where simulations have shown that the bit error rate can be significantly reduced compared to using common suboptimal algorithms. Second, an MIQP algorithm tailored for MPC is presented. The algorithm uses a branch and bound method where the relaxed node problems are solved by a dual active set QP algorithm. In this QP algorithm, the KKT systems are solved using Riccati recursions in order to decrease the computational complexity. Simulation results show that both the proposed QP solver and MIQP solver have lower computational complexity compared to corresponding generic solvers. Third, the dual active set QP algorithm is enhanced using ideas from gradient projection methods. The performance of this enhanced algorithm is shown to be comparable with the existing commercial state-of-the-art QP solver \cplex for some random linear MPC problems. Fourth, an algorithm for efficient computation of the search directions in an SDP solver for a proposed alternative SDP relaxation applicable to MPC problems with binary control signals is presented. The SDP relaxation considered has the potential to give a tighter lower bound on the optimal objective function value compared to the QP relaxation that is traditionally used in branch and bound for these problems, and its computational performance is better than the ordinary SDP relaxation for the problem. Furthermore, the tightness of the different relaxations is investigated both theoretically and in numerical experiments.

Place, publisher, year, edition, pages
Institutionen för systemteknik, 2008. 231 p.
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1158
Integer Quadratic Programming, Model Predictive Control, Hybrid Systems, Semidefinite Programming, Code Division Multiple Access, Multiuser Detection, Automatic Control, Communication
National Category
Control Engineering
urn:nbn:se:liu:diva-10642 (URN)978-91-85523-03-0 (ISBN)
Public defence
2008-02-27, Visionen, Hus C, Linköpings universitet, Linköping, 10:15 (English)
This material is posted here with permission of the IEEE. Such permission of the IEEE does not in any way imply IEEE endorsement of any of the Linköping University's products or services. Internal or personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution must be obtained from the IEEE by writing to By choosing to view this material, you agree to all provisions of the copyright laws protecting it.Available from: 2008-03-18 Created: 2008-03-18 Last updated: 2016-08-31

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