Utilizing low rank properties when solving KYP-SDPs

Harju, J.; Wallin, Ragnar; Hansson, Anders

doi:10.1109/cdc.2006.376704

Cited by 4 publications

(2 citation statements)

References 21 publications

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“…By using the results in [24], (30)- (32) can be solved at a total cost of O(n 3 ). Finally, the dual variables ∆Z i in (29) are easily obtained by matrix inversions.…”

Section: Preconditioner IImentioning

confidence: 99%

A tailored inexact interior-point method for systems analysis

Johansson

Hansson

2008

2008 47th IEEE Conference on Decision and Control

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Within the area of systems analysis there are several problem formulations that can be rewritten as semidefinite programs. Increasing demand on computational efficiency and ability to solve large scale problems make the available generic solvers inadequate. In this paper structure knowledge is utilized to derive tailored calculations and to incorporate adaptation to the different properties that appear in a proposed inexact interior-point method. Abstract-Within the area of systems analysis there are several problem formulations that can be rewritten as semidefinite programs. Increasing demand on computational efficiency and ability to solve large scale problems make the available generic solvers inadequate. In this paper structure knowledge is utilized to derive tailored calculations and to incorporate adaptation to the different properties that appear in a proposed inexact interior-point method. KeywordsI. INTRODUCTION In this paper a structured semidefinite programming (SDP) problem is defined and a tailored algorithm is proposed and evaluated. The problem formulation, can for example be applied to analysis of polytopic linear differential inclusions (LDIs). The reformulation from systems analysis problems to SDPs is described in [9] These solvers solve the optimization problem on a general form. The problem size will increase with the number of constraints and the number of matrix variables. Hence, for large scale problems, generic solvers will not have an acceptable solution time or terminate within an acceptable number of function calls. It is necessary to utilize the problem structure to speed up the performance. Here an algorithm is described that uses inexact serch directions for an infeasible interior-point method. A memory efficient iterative solver is used to compute the search directions in each step of the interior-point method. In each step of the algorithm, the error tolerance for the iterative solver decreases, and hence the initial steps are less expensive to calculate than the last ones.Iterative solvers for linear systems of equations are well studied in the literature. For applications to optimization and preconditioning for interior-point methods see [12]

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“…By using the results in [24], (30)- (32) can be solved at a total cost of O(n 3 ). Finally, the dual variables ∆Z i in (29) are easily obtained by matrix inversions.…”

Section: Preconditioner IImentioning

confidence: 99%

A tailored inexact interior-point method for systems analysis

Johansson

Hansson

2008

2008 47th IEEE Conference on Decision and Control

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“…These include cutting-plane methods that avoid the introduction of the matrix variable P by working directly with the frequency-domain inequality (2) [Par99, KMJ00, KM01, KMJ03, WKH08], interior-point methods based on new barrier functions for the convex set defined by the frequency domain inequalities [KM01, KM03,KM07], and customized implementations of standard interior-point methods for semidefinite programming [HV00, AV02, GHNV03, Hac03, GH03, VBW + 05, LP04,RV06,HWH06,LV07].…”

Section: Introductionmentioning

confidence: 99%

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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