A note on properties of condition numbers

Gonzaga, Clóvis C.; Lara, Hugo

doi:10.1016/s0024-3795(96)00409-0

Cited by 16 publications

(21 citation statements)

References 4 publications

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“…Observe that this parameter is invariant when A is scaled by a constant. (See Gonzaga and Lara, 1996, for the latest result on this quantity.) Thus, the total number of interior-point-method iterations is bounded by O(͉B͉ ͉N͉n It should be noted that this complexity bound does not depend on the value of Ȑ given to Algorithm LIP at the start of the first main loop iteration.…”

Section: The Layered Interior Point Algorithmmentioning

confidence: 96%

How Partial Knowledge Helps to Solve Linear Programs

1996

Journal of Complexity

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We present results on how partial knowledge helps to solve linear programs. In particular, if a linear system, Ax ϭ b and x Ն 0, has an interior feasible point, then we show that finding a feasible point to this system can be done in O(n 2.5 c(A)) iterations by the layered interior-point method, and each iteration solves a leastsquares problem, where n is the dimension of vector x and c(A) is the condition number of matrix A defined by Vavasis and Ye. This complexity bound is reduced by a factor n from that when this property does not exists. We also present a result for solving the problem using a little strong knowledge.

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Section: The Layered Interior Point Algorithmmentioning

confidence: 96%

How Partial Knowledge Helps to Solve Linear Programs

1996

Journal of Complexity

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“…For instance, (C * AC) † is unbounded if R(C * AC) is not closed; or C(C * AC) † C * A may have range strictly contained in S. However, the wide range of applications of the right side of formula (4) makes it desirable to establish its exact relationship with P A,S . In fact, projectors like C(C * AC) † C * A appear explicitly in papers on scaled projections [53], [46], [34], [31], [60], [14], linear least squares problems [28], [29], linear feasibility [28], [29], [17], signal processing [36], [10], [58] and so on. A first observation is that one needs to verify if R(C * AC) is closed.…”

Section: (A S)mentioning

confidence: 99%

A classification of projectors

Corach

Maestripieri²,

Stojanoff

2005

Topological Algebras, Their Applications, and Related Topics

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A positive operator A and a closed subspace S of a Hilbert space H are called compatible if there exists a projector Q onto S such that AQ = Q * A. Compatibility is shown to depend on the existence of certain decompositions of H and the ranges of A and A 1/2 . It also depends on a certain angle between A(S) and the orthogonal of S.2000 Mathematics Subject Classification: 47A64, 47A07, 46C99. Key words and phrases: oblique projectors, least square problems, scaled projectors, positive operators.Research partially supported by CONICET (PIP 2083/02), Universidad de Buenos Aires (UBACYT X050), Universidad de La Plata 11 X350 and ANPCYT (PICT03-9521).The paper is in final form and no version of it will be published elsewhere.[145] 146G. CORACH ET AL.For each closed subspace S of H let Q S denote the set of all projectors with range S. For each (bounded linear semidefinite) positive operator A on H consider the set P(A, S) = {Q ∈ Q S : AQ = Q * A}, i.e., all Q with range S which are Hermitian with respect to the sesquilinear form ξ, η A = Aξ, η . Of course, P(A, S) can be empty (see examples below); we say that A, S are compatible if P(A, S) is not empty. This condition can be read in terms of different space decompositions, range inclusions and angles between certain closed subspaces of H. It is known [19] that, if A and S are compatible then a distinguished element P A,S of P(A, S) exists which has optimal properties. We show explicit formulas for P A,S which are computationally useful.Many results on oblique projectors can be found in the papers by Afriat [1], Davis [22], Ljance [43], Mizel and Rao [45], Halmos [33], Greville [32], Gerisch [30], Pták [49]. Projectors which are Hermitian with respect to a positive matrix have been studied by Mitra and Rao [44] and Baksalary and Kala [9]. More recently, Hassi and Nordstrom [35]studied projectors which are Hermitian with respect to a self-adjoint operator but with emphasis on the case in which P(A, S) is a singleton. In [47], Pasternak-Winiarski studied the analyticity of the map A → P A,S , where A runs over the set of positive invertible operators. The map (A, S) → P A,S is studied by Andruchow, Corach and Stojanoff [6], for positive invertible A. For general selfadjoint A, several results on P(A, S) can be found in [19] and the present paper can be seen as its continuation. Additional results by the authors are contained in [20] and [21]. The latter makes a link between oblique projectors and abstract splines in the sense of Atteia [8]. It is natural that this type of least square approximation results appears in this context, because P A,S is a kind of orthogonal projector for an appropriate inner product. In particular, oblique projectors, mainly in the finite-dimensional setting, appear frequently under the form of "scaled projectors", i.e., projectors which are Hermitian with respect to a positive diagonal matrix. The reader is referred to the papers by Stewart [53], O'Leary [46], Hanke and Neumann [34], Gonzaga and Lara [31], Wei [60], Forsgren [28], Vavasis [1...

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“…The numberχ A was first introduced implicitly by Dikin [2] in the study of primal affine scaling (AS) algorithms, and was later studied by several researchers including Vanderbei and Lagarias [27], Todd [21], and Stewart [18]. Properties ofχ A are studied in [3,25,26].…”

Section: Introductionmentioning

confidence: 99%

“…A.The rows of A are linearly independent. Under the above assumptions, it is well known that for any ν > 0 the system, xs = νe,(3)…”

mentioning

confidence: 99%

A Polynomial Predictor-Corrector Trust-Region Algorithm for Linear Programming

Lan¹,

Monteiro²,

Tsuchiya³

2009

SIAM J. Optim.

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In this paper we present a scaling-invariant, interior-point, predictor-corrector type algorithm for linear programming (LP) whose iteration-complexity is polynomially bounded by the dimension and the logarithm of a certain condition number of the LP constraint matrix. At the predictor stage, the algorithm either takes the step along the standard affine scaling (AS) direction or a new trust-region type direction, whose construction depends on a scaling-invariant bipartition of the variables determined by the AS direction. This contrasts with the layered least squares direction introduced in S. Vavasis and Y. Ye [Math. Program., 74 (1996), pp. 79-120], whose construction depends on multiple-layered partitions of the variables that are not scaling-invariant. Moreover, it is shown that the overall arithmetic complexity of the algorithm (weakly) depends on the right-hand side and the cost of the LP in view of the work involved in the computation of the trust region steps.

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A note on properties of condition numbers

Cited by 16 publications

References 4 publications

How Partial Knowledge Helps to Solve Linear Programs

How Partial Knowledge Helps to Solve Linear Programs

A classification of projectors

A Polynomial Predictor-Corrector Trust-Region Algorithm for Linear Programming

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