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Relaxations Applicable to Mixed Integer Predictive Control – Comparisons and Efficient Computations
Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
University of Californa, LA, USA.
2007 (English)In: Proceedings of the 46th IEEE Conference on Decision and Control, 2007, 4103-4109 p.Conference paper (Refereed)
Abstract [en]

In this work, different relaxations applicable to an MPC problem with a mix of real valued and binary valued control signals are compared. In the problem description considered, there are linear inequality constraints on states and control signals. The relaxations are related theoretically and both the tightness of the bounds and the computational complexities are compared in numerical experiments. The relaxations considered are the quadratic programming (QP) relaxation, the standard semidefinite programming (SDP) relaxation and an equality constrained SDP relaxation. The result is that the standard SDP relaxation is the one that usually gives the best bound and is most computationally demanding, while the QP relaxation is the one that gives the worst bound and is least computationally demanding. The equality constrained relaxation presented in this paper often gives a better bound than the QP relaxation and is less computationally demanding compared to the standard SDP relaxation. Furthermore, it is also shown how the equality constrained SDP relaxation can be efficiently computed by solving the Newton system in an Interior Point algorithm using a Riccati recursion. This makes it possible to compute the equality constrained relaxation with approximately linear computational complexity in the prediction horizon.

Place, publisher, year, edition, pages
2007. 4103-4109 p.
Keyword [en]
Newton method, Riccati equations, Computational complexity, Predictive control, Quadratic programming, Relaxation theory, Interior point algorithm, Newton system, Riccati recursion, Linear computational complexity, Linear inequality constraints, Mixed integer predictive control
National Category
Engineering and Technology Control Engineering
URN: urn:nbn:se:liu:diva-12907DOI: 10.1109/CDC.2007.4434608ISBN: 978-1-4244-1497-0ISBN: 978-1-4244-1498-7OAI: diva2:17357
46th IEEE Conference on Decision and Control, New Orleans, LA, USA, December, 2007
Available from: 2008-03-18 Created: 2008-03-18 Last updated: 2013-02-27Bibliographically approved
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: 2009-04-22

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