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A Preprocessing Algorithm for MIQP Solvers with Applications to MPC
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.
2004 (English)In: Proceedings of the 43rd IEEE Conference on Decision and Control, 2004, 2497-2502 p.Conference paper (Refereed)
Abstract [en]

In this paper a preprocessing algorithm for unconstrained mixed integer quadratic programming problems and binary quadratic programming problems is presented. The algorithm applies to problems with certain properties, which are further described in the paper. When the algorithm is applied to a problem with these properties, the optimal value for some or all integer variables can be computed without approximations in polynomial time. The algorithm is first derived for the binary quadratic programming problem and the resultis then extended to the mixed integer quadratic programming problem by transforming the latter problem into the first problem. Both mentioned quadratic programming problems have several important applications. In this paper, the focus is on model predictive control problems with both real-valued and binary control signals. As an illustration of the method, the algorithm is applied to two different problems of this type.

Place, publisher, year, edition, pages
2004. 2497-2502 p.
Keyword [en]
Predictive control, Integer programming, Quadratic programming
National Category
Engineering and Technology Control Engineering
URN: urn:nbn:se:liu:diva-12902DOI: 10.1109/CDC.2004.1428790ISBN: 0-7803-8682-5OAI: diva2:17352
43rd IEEE Conference on Decision and Control, Paradise Island, Bahamas, December, 2004
Available from: 2008-03-18 Created: 2008-03-18 Last updated: 2013-03-26
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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