What are Schur complements, anyway?

Carlson, David

doi:10.1016/0024-3795(86)90127-8

Cited by 106 publications

(47 citation statements)

References 18 publications

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“…For further details on the generalized Schur complement, see Carlson, Haynsworth and Markham [6] and Ando [1]. Carlson [5] gives an interesting survey of results on both classical and generalized Schur complements, along with an extensive bibliography.…”

Section: 4mentioning

confidence: 99%

Inertia-Controlling Methods for General Quadratic Programming

et al. 1991

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Abstract. Active-set quadratic programming (QP) methods use a working set to define the search direction and multiplier estimates. In the method proposed by Fletcher in 1971, and in several subsequent mathematically equivalent methods, the working set is chosen to control the inertia of the reduced Hessian, which is never permitted to have more than one nonpositive eigenvalue. (We call such methods inertia-controlling.) This paper presents an overview of a generic inertia-controlling QP method, including the equations satisfied by the search direction when the reduced Hessian is positive definite, singular and indefinite. Recurrence relations are derived that define the search direction and Lagrange multiplier vector through equations related to the Karush-Kuhn-Tucker system. We also discuss connections with inertia-controlling methods that maintain an explicit factorization of the reduced Hessian matrix.

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Section: 4mentioning

confidence: 99%

Inertia-Controlling Methods for General Quadratic Programming

et al. 1991

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“…If D is not a square matrix then a pseudo-Schur complement of D in M can still be defined [7,8]. Now, considering the matrices K andM partitioned as follows…”

Section: ẋ(T) = a X(t) + B U(t) Y(t) = C X(t)mentioning

confidence: 99%

Model reduction in large scale MIMO dynamical systems via the block Lanczos method

Heyouni¹,

Jbilou²,

Messaoudi³

et al. 2008

Comput.appl.math.

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“…Hence, ad hoc evolutionary and enumeration-tree searches are used such as those found in the works of Bagajewicz 12 and Carnero et al 13,14 as well as in that of Gala and Bagajewicz 20 who employ cutsets. As will be shown, these ad hoc approaches are not necessary in solving the estimability problem when the variable classification method of Kelly 1 and the Schur complement [21][22][23] are used. In addition, when optimizing cost subject to precision or variability constraints is the goal such as in the works of Bagajewicz and Cabrera 16 and Chmielewski et al, 17 sophisticated MILP and linear matrix inequality formulations are used to parameterize the diagonal of the covariance matrices as functions of the sensor location logic variables (measured vs. unmeasured).…”

Section: Introductionmentioning

confidence: 98%

A new and improved MILP formulation to optimize observability, redundancy and precision for sensor network problems

Kelly

Zyngier

2008

AIChE Journal

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in Wiley InterScience (www.interscience.wiley.com).New mathematical formulations of observability, redundancy, and an improved formulation for precision are provided which can be explicitly and analytically solved using mixed integer linear programming (MILP). By using the Schur complement found at the heart of both Gaussian elimination and Cholesky factorization for direct block matrix reduction and the variable classification and covariance calculations found in the reconciliation, regression, and regularization approach of Kelly, it is possible to efficiently optimize the overall instrumentation cost considering both estimability and variability as constraints during the branch-and-bound search of the MILP. Two illustrative examples are highlighted which minimize the cost of sensor placement subject to software and hardware redundancy of the measured variables, observability of the unmeasured variables, and their precision (i.e., inverse of their variance). This formulation is well suited to the problems of designing as well as retrofitting sensor locations in arbitrary networks.

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What are Schur complements, anyway?

Cited by 106 publications

References 18 publications

Inertia-Controlling Methods for General Quadratic Programming

Inertia-Controlling Methods for General Quadratic Programming

Model reduction in large scale MIMO dynamical systems via the block Lanczos method

A new and improved MILP formulation to optimize observability, redundancy and precision for sensor network problems

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