A Structure Exploiting Preprocessor for Semidefinite Programs Derived From the Kalman-Yakubovich-Popov Lemma

Wallin, Ragnar; Hansson, Anders; Johansson, Janne Harju

doi:10.1109/tac.2009.2014922

Cited by 17 publications

(22 citation statements)

References 20 publications

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“…For the first one of these the problem is just solved in a straightforward way using the solver SDPT3. The second one is using the preprocessing of STRUL in conjunction with SDPT3, and the third one is using the software KYPD, [WHJ09], in conjunction with SDPT3.…”

Section: Integral Quadratic Constraintsmentioning

confidence: 99%

“…In this paper we will focus on the discrete-time case. All the other cases can be transformed to this case, see [WHJ09].) By applying the KYP lemma the optimization problem in (5) is reformulated as an optimization problem of the form minimize c T x subject to A B E 0…”

Section: Introductionmentioning

confidence: 99%

“…It is therefore of interest to explore reformulations of Popov function constraints that result in problems that can be solved by general-purpose solvers at a cost that is comparable to the dedicated solvers listed above. (An example of such a preprocessor, based on a reformulation of the dual problem, is discussed in [WHJ09].) In particular, several general-purpose packages now exploit low-rank structure in the coefficient matrices of SDPs, in addition to sparsity.…”

Section: Introductionmentioning

confidence: 99%

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Sampling method for semidefinite programmes with non-negative Popov function constraints

Hansson

Vandenberghe

2013

International Journal of Control

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An important class of optimisation problems in control and signal processing involves the constraint that a Popov function is non-negative on the unit circle or the imaginary axis. Such a constraint is convex in the coefficients of the Popov function. It can be converted to a finite-dimensional linear matrix inequality via the Kalman-Yakubovich-Popov lemma. However, the linear matrix inequality reformulation requires an auxiliary matrix variable and often results in a very large semidefinite programming problem. Several recently published methods exploit problem structure in these semidefinite programmes to alleviate the computational cost associated with the large matrix variable. These algorithms are capable of solving much larger problems than general-purpose semidefinite programming packages. In this paper, we address the same problem by presenting an alternative to the linear matrix inequality formulation of the non-negative Popov function constraint. We sample the constraint to obtain an equivalent set of inequalities of low dimension, thus avoiding the large matrix variable in the linear matrix inequality formulation. Moreover, the resulting semidefinite programme has constraints with low-rank structure, which allows the problems to be solved efficiently by existing semidefinite programming packages. The sampling formulation is obtained by first expressing the Popov function inequality as a sum-of-squares condition imposed on a polynomial matrix and then converting the constraint into an equivalent finite set of interpolation constraints. A complexity analysis and numerical examples are provided to demonstrate the performance improvement over existing techniques

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Section: Integral Quadratic Constraintsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Sampling method for semidefinite programmes with non-negative Popov function constraints

Hansson

Vandenberghe

2013

International Journal of Control

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“…This is mainly due to insufficient memory or processing power of available centralized work stations or in general our limited capability to solve the centralized problem in a timely manner. Such challenges are commonly addressed by exploiting structure in the problem and devising efficient solvers that use the available computational and memory resources more wisely, e.g., see Wallin et al (2009);Hansson and Vandenberghe (2000); Rice and Verhaegen (2009);Massioni and Verhaegen (2009).…”

Section: Introductionmentioning

confidence: 99%

Divide and Conquer: Distributed Optimization and Robustness Analysis

Pakazad

2015

Linköping Studies in Science and Technology. Dissertations

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Linköping 2015Cover illustration: A design approach for distributed optimization alogirhtms, used in papers C and F. It states that given the sparstiy graph of a coupled optimization problem, defined in Section 5.1, we can group its variables and its constituent subprobelms so that the coupling structure can be represented using a tree. We can then solve the problem distributedly using message passing, which utilizes the tree as its computational graph. To my parents AbstractAs control of large-scale complex systems has become more and more prevalent within control, so has the need for analyzing such controlled systems. This is particularly due to the fact that many of the control design approaches tend to neglect intricacies in such systems, e.g., uncertainties, time delays, nonlinearities, so as to simplify the design procedure. Robustness analysis techniques allow us to assess the effect of such neglected intricacies on performance and stability. Performing robustness analysis commonly requires solving an optimization problem. However, the number of variables of this optimization problem, and hence the computational time, scales badly with the dimension of the system. This limits our ability to analyze large-scale complex systems in a centralized manner. In addition, certain structural constraints, such as privacy requirements or geographical separation, can prevent us from even forming the analysis problem in a centralized manner.In this thesis, we address these issues by exploiting structures that are common in large-scale systems and/or their corresponding analysis problems. This enables us to reduce the computational cost of solving these problems both in a centralized and distributed manner. In order to facilitate distributed solutions, we employ or design tailored distributed optimization techniques. Particularly, we propose three distributed optimization algorithms for solving the analysis problem, which provide superior convergence and/or computational properties over existing algorithms. Furthermore, these algorithms can also be used for solving general loosely coupled optimization problems that appear in a variety of fields ranging from control, estimation and communication systems to supply chain management and economics.v Populärvetenskaplig sammanfattningRegulatorer för styrning av system är ofta designade med hjälp av en beskrivning, typiskt en matematisk modell, av det man vill styra. Dessa modeller är oftast endast approximativa beskrivningar av det verkliga systemet och därför behäftade med osäkerheter. Detta medför att det beteende man förväntar sig av ett reglerat system baserat på den matematiska modellen inte alltid stämmer överens med det observerade beteendet. Att analysera hur stor denna avvikelse kan bli kallas robusthetsanalys. Detta är ett väl studerat område för mindre och medelstora system. Dock är kunskaperna begränsade för riktigt stora system, speciellt om man inte vill göra en alltför konservativ analys. Exempel på stora system är kraftnät och flygplan. Kraftnät är stora på gr...

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“…Because of this fact, tailor-made solvers for various types of problems have been developed, for example for programs derived from the Kalman-Yakubovic-Popov lemma [10], [11], [12], [13], [14].…”

Section: Introductionmentioning

confidence: 99%

Low-rank exploitation in semidefinite programming for control

Falkeborn

Löfberg

Hansson

2011

International Journal of Control

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Many control related problems can be cast as semidefinite programs but, even though there exist polynomial time algorithms and good publicly available solvers, the time it takes to solve these problems can be long. Something many of these problems have in common, is that some of the variables enter as matrix valued variables. This leads to a low-rank structure in the basis matrices which can be exploited when forming the Newton equations. In this paper, we describe the how this can be done, and show how our code can be used when using SDPT3. The idea behind this is old and is implemented in LMI Lab, but we show that when using a modern algorithm, the computational time can be reduced. Finally, we describe how the modeling language YALMIP is changed in such a way that our code can be interfaced using standard YALMIP commands, which greatly simplifies for the user. Keywords: Semidefinite programming, Structure exploitationLowrank exploitation in semidefinite programming for control Rikard Falkeborn, Johan Löfberg and Anders HanssonAbstract-Many control related problems can be cast as semidefinite programs but, even though there exist polynomial time algorithms and good publicly available solvers, the time it takes to solve these problems can be long. Something many of these problems have in common, is that some of the variables enter as matrix valued variables. This leads to a low-rank structure in the basis matrices which can be exploited when forming the Newton equations. In this paper, we describe the how this can be done, and show how our code can be used when using SDPT3. The idea behind this is old and is implemented in LMI Lab, but we show that when using a modern algorithm, the computational time can be reduced. Finally, we describe how the modeling language YALMIP is changed in such a way that our code can be interfaced using standard YALMIP commands, which greatly simplifies for the user.

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A Structure Exploiting Preprocessor for Semidefinite Programs Derived From the Kalman-Yakubovich-Popov Lemma

Cited by 17 publications

References 20 publications

Sampling method for semidefinite programmes with non-negative Popov function constraints

Sampling method for semidefinite programmes with non-negative Popov function constraints

Divide and Conquer: Distributed Optimization and Robustness Analysis

Low-rank exploitation in semidefinite programming for control

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