Structure exploitation in Semi-Definite Programs for Systems Analysis

Johansson, Janne Harju; Hansson, Anders

doi:10.3182/20080706-5-kr-1001.01700

Cited by 1 publication

(2 citation statements)

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“…This results in a preconditioner that can be condensed to solving a linear system of equations of the same size as if there were only one constraint in the optimization problem with a simple scaling matrix. For a thorough description and simulation results, see [25], where it was noted that the described assumption is only valid in the initial steps of the algorithm. An explanation is that when the iterates tend to the boundary of the feasible region the eigenvalues of W i for the active constraint are not close to each other.…”

Section: Preconditioner Imentioning

confidence: 99%

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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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Section: Preconditioner Imentioning

confidence: 99%

“…In [25] some preliminary results were presented. However, it was noted that the convergence of the iterative solver was only satisfactory initially in the algorithm and hence further work was needed to cover a larger class of problems.…”

Section: Introductionmentioning

confidence: 99%