| 1. |
- Grussler, Christian, et al.
(författare)
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A Symmetry Approach for Balanced Truncation of Positive Linear Systems
- 2012
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Ingår i: IEEE Control Systems Society Conference 2012, Maui. - 0191-2216. ; , s. 4308-4313
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Konferensbidrag (refereegranskat)abstract
- We consider model order reduction of positive linear systems and show how a symmetry characterization can be used in order to preserve positivity in balanced truncation. The reduced model has the additional feature of being symmetric.
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| 2. |
- Grussler, Christian, et al.
(författare)
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Efficient Proximal Mapping Computation for Low-Rank Inducing Norms
- 2022
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Ingår i: Journal of Optimization Theory and Applications. - : Springer Science and Business Media LLC. - 0022-3239 .- 1573-2878. ; 192:1, s. 168-194
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Tidskriftsartikel (refereegranskat)abstract
- Low-rank inducing unitarily invariant norms have been introduced to convexify problems with a low-rank/sparsity constraint. The most well-known member of this family is the so-called nuclear norm. To solve optimization problems involving such norms with proximal splitting methods, efficient ways of evaluating the proximal mapping of the low-rank inducing norms are needed. This is known for the nuclear norm, but not for most other members of the low-rank inducing family. This work supplies a framework that reduces the proximal mapping evaluation into a nested binary search, in which each iteration requires the solution of a much simpler problem. The simpler problem can often be solved analytically as demonstrated for the so-called low-rank inducing Frobenius and spectral norms. The framework also allows to compute the proximal mapping of increasing convex functions composed with these norms as well as projections onto their epigraphs.
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| 3. |
- Grussler, Christian, et al.
(författare)
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Identification of externally positive systems
- 2018
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Ingår i: 2017 IEEE 56th Annual Conference on Decision and Control, CDC 2017. - 9781509028733 ; 2018-January, s. 6549-6554
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Konferensbidrag (refereegranskat)abstract
- We consider identification of externally positive linear discrete-time systems from input/output data. The proposed method is formulated as a semidefinite program, and is guaranteed to identify models that are ellipsoidal cone-invariant and, consequently, externally positive. We demonstrate empirically that this cone-invariance approach can significantly reduce the conservatism associated with methods that enforce internal positivity as a sufficient condition for external positivity.
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| 4. |
- Grussler, Christian, et al.
(författare)
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Local convergence of proximal splitting methods for rank constrained problems
- 2018
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Ingår i: 2017 IEEE 56th Annual Conference on Decision and Control, CDC 2017. - 9781509028733 ; 2018-January, s. 702-708
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Konferensbidrag (refereegranskat)abstract
- We analyze the local convergence of proximal splitting algorithms to solve optimization problems that are convex besides a rank constraint. For this, we show conditions under which the proximal operator of a function involving the rank constraint is locally identical to the proximal operator of its convex envelope, hence implying local convergence. The conditions imply that the non-convex algorithms locally converge to a solution whenever a convex relaxation involving the convex envelope can be expected to solve the non-convex problem.
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| 5. |
- Grussler, Christian, et al.
(författare)
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Low-rank inducing norms with optimality interpretations∗
- 2018
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Ingår i: SIAM Journal on Optimization. - 1052-6234. ; 28:4, s. 3057-3078
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Tidskriftsartikel (refereegranskat)abstract
- Optimization problems with rank constraints appear in many diverse fields such as control, machine learning, and image analysis. Since the rank constraint is nonconvex, these problems are often approximately solved via convex relaxations. Nuclear norm regularization is the prevailing convexifying technique for dealing with these types of problem. This paper introduces a family of low-rank inducing norms and regularizers which include the nuclear norm as a special case. A posteriori guarantees on solving an underlying rank constrained optimization problem with these convex relaxations are provided. We evaluate the performance of the low-rank inducing norms on three matrix completion problems. In all examples, the nuclear norm heuristic is outperformed by convex relaxations based on other low-rank inducing norms. For two of the problems there exist low-rank inducing norms that succeed in recovering the partially unknown matrix, while the nuclear norm fails. These low-rank inducing norms are shown to be representable as semidefinite programs. Moreover, these norms have cheaply computable proximal mappings, which make it possible to also solve problems of large size using first-order methods.
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| 6. |
- Grussler, Christian, et al.
(författare)
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Low-Rank Optimization with Convex Constraints
- 2018
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Ingår i: IEEE Transactions on Automatic Control. - 0018-9286. ; 63:11, s. 4000-4007
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Tidskriftsartikel (refereegranskat)abstract
- The problem of low-rank approximation with convex constraints, which appears in data analysis, system identification, model order reduction, low-order controller design and low-complexity modelling is considered. Given a matrix, the objective is to find a low-rank approximation that meets rank and convex constraints, while minimizing the distance to the matrix in the squared Frobenius norm. In many situations, this non-convex problem is convexified by nuclear norm regularization. However, we will see that the approximations obtained by this method may be far from optimal. Here, we propose an alternative convex relaxation that uses the convex envelope of the squared Frobenius norm and the rank constraint. With this approach, easily verifiable conditions are obtained under which the solutions to the convex relaxation and the original non-convex problem coincide. An SDP representation of the convex envelope is derived, which allows us to treat several known problems. Our example on optimal low-rank Hankel approximation/model reduction illustrates that the proposed convex relaxation performs consistently better than nuclear norm regularization as well as balanced truncation.
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| 7. |
- Grussler, Christian, et al.
(författare)
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Modified balanced truncation preserving ellipsoidal cone-invariance
- 2014
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Ingår i: IEEE Xplore Digital Library. ; , s. 2365-2370
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Konferensbidrag (övrigt vetenskapligt/konstnärligt)abstract
- We consider model order reduction of stable linear systems which leave ellipsoidal cones invariant. We show how balanced truncation can be modified to preserve cone- invariance. Additionally, this implies a method to perform external positivity preserving model reduction for a large class of systems.
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| 8. |
- Grussler, Christian, et al.
(författare)
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On optimal low-rank approximation of non-negative matrices
- 2015
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Ingår i: 2015 IEEE 54th Annual Conference on Decision and Control (CDC). - 9781479978861 ; , s. 5278-5283
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Konferensbidrag (refereegranskat)abstract
- For low-rank Frobenius-norm approximations of matrices with non-negative entries, it is shown that the Lagrange dual is computable by semi-definite programming. Under certain assumptions the duality gap is zero. Even when the duality gap is non-zero, several new insights are provided.
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| 9. |
- Grussler, Christian, et al.
(författare)
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On second-order cone positive systems
- 2021
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Ingår i: SIAM Journal on Control and Optimization. - 0363-0129. ; 59:4, s. 2717-2739
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Tidskriftsartikel (refereegranskat)abstract
- Internal positivity offers a computationally cheap certificate for external (inputoutput) positivity of a linear time-invariant system. However, the drawback with this certificate lies in its realization dependency. First, computing such a realization requires finding a polyhedral cone with a potentially high number of extremal generators that lifts the dimension of the state-space representation, significantly. Second, not all externally positive systems possess an internally positive realization. Third, in many typical applications such as controller design, system identification, and model order reduction, internal positivity is not preserved. To overcome these drawbacks, we present a tractable sufficient certificate of external positivity based on second-order cones. This certificate does not require any special state-space realization: if it succeeds with a possibly non-minimal realization, then it will do so with any minimal realization. While there exist systems where this certificate is also necessary, we also demonstrate how to construct systems, where both second-order and polyhedral cones as well as other certificates fail. Nonetheless, in contrast to other realization independent certificates, the second-order-cone one appears to be favorable in terms of applicability and conservatism. Three applications are representatively discussed to underline its potential. We show how the certificate can be used to find externally positive approximations of nearly externally positive systems and demonstrate that this may help to reduce system identification errors. The same algorithm is used then to design state-feedback controllers that provide closed-loop external positivity, a common approach to avoid over- and undershooting of the step response. Last, we present modifications to generalized balanced truncation such that external positivity is preserved for those systems, where our certificate applies.
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| 10. |
- Grussler, Christian, et al.
(författare)
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Optimality interpretations for atomic norms
- 2019
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Ingår i: 2019 18th European Control Conference, ECC 2019. - 9783907144008 ; , s. 1473-1477
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Konferensbidrag (refereegranskat)abstract
- Atomic norms occur frequently in data science and engineering problems such as matrix completion, sparse linear regression, system identification and many more. These norms are often used to convexify non-convex optimization problems, which are convex apart from the solution lying in a non-convex set of so-called atoms. For the convex part being a linear constraint, the ability of several atomic norms to solve the original non-convex problem has been analyzed by means of tangent cones. This paper presents an alternative route for this analysis by showing that atomic norm convexifcations always provide an optimal convex relaxation for some related non-convex problems. As a result, we obtain the following benefits: (i) treatment of arbitrary convex constraints, (ii) potentially obtaining solutions to the non-convex problem with a posteriori success certificates, (iii) utilization of additional prior knowledge through the design or learning of the non-convex problem.
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