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Linear complementarity, linear and nonlinear programming
01 Jan 1988-
About: The article was published on 1988-01-01 and is currently open access. It has received 1012 citations till now. The article focuses on the topics: Mixed complementarity problem & Complementarity theory.
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01 Jan 1992
4 citations
TL;DR: In this paper, two new classes of matrices that contain the class of P-matrices and can be recognized in polynomial time were introduced, which are related to the known classes of Hidden Minkowski and hidden positive row diagonally dominant matrices.
4 citations
TL;DR: In this article, it was shown that Hager's condition number estimator is equivalent to the conditional gradient algorithm applied to the problem of maximizing the 1-norm of a matrix-vector product over the unit sphere in the 1 norm.
Abstract: Techniques for estimating the condition number of a nonsingular matrix are developed. It is shown that Hager’s 1-norm condition number estimator is equivalent to the conditional gradient algorithm applied to the problem of maximizing the 1-norm of a matrix-vector product over the unit sphere in the 1-norm. By changing the constraint in this optimization problem from the unit sphere to the unit simplex, a new formulation is obtained which is the basis for both conditional gradient and projected gradient algorithms. In the test problems, the spectral projected gradient algorithm yields condition number estimates at least as good as those obtained by the previous approach. Moreover, in some cases, the spectral gradient projection algorithm, with a careful choice of the parameters, yields improved condition number estimates.
4 citations
Cites background from "Linear complementarity, linear and ..."
...Pentadiagonal matrices (Murty, 1988) with mii = 6 mi,i−1 = mi−1,i = −4 mi,i−2 = mi−2,i = 1 and all other elements equal to zero....
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...Type V. Murty matrices (Murty, 1988) with the structure 1 2 1 2 2 1 ....
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...Type VI. Fathy matrices (Murty, 1988) of the form F = MT M where M is a Murty matrix....
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...Murty matrices (Murty, 1988) with the structure 1 2 1 2 2 1 ....
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...Pentadiagonal matrices (Murty, 1988) with mii = 6 mi,i−1 = mi−1,i = −4 mi,i−2 = mi−2,i = 1 and all other elements equal to zero....
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TL;DR: In this paper, the authors consider the problem of studying specified perturbations of the data such that for a given optimization problem some optimality criterion remains satisfied, i.e.
Abstract: When we speak about parametric programming, sensitivity analysis or related topics, we usually mean the problem of studying specified perturbations of the data such that for a given optimization problem some optimality criterion remains satisfied. In this paper, we turn to another question. Suppose that A is a matrix having a specific property P. What are the maximal allowable variations of the data such that the property still remains valid for the matrix? We study two basic forms of perturbations. The first is a perturbation in a given direction, which is closely related to parametric programming. The second type consists of all possible data variations in a neighbourhood specified by a certain matrix norm; this is related to the tolerance approach to sensitivity analysis or to stability. The matrix properties discussed in this paper are positive definiteness; M-matrix, H-matrix and P-matrix property; total positivity; inverse M-matrix property and inverse nonnegativity.
4 citations
01 Jan 2011
TL;DR: In this article, generalized monotone maps are used in the analysis and solution of variational inequality and complementarity problems, and affine pseudomonotone mapping, affine quasimonotone map, generalized positive-subdefinite matrices, and the linear complementarity problem.
Abstract: In this chapter, we present some classes of generalized monotone maps and their relationship with the corresponding concepts of generalized convexity. We present results of generalized monotone maps that are used in the analysis and solution of variational inequality and complementarity problems. In addition, we obtain various characterizations and establish a connection between affine pseudomonotone mapping, affine quasimonotone mapping, positive-subdefinite matrices, generalized positive-subdefinite matrices, and the linear complementarity problem. These characterizations are useful for extending the applicability of Lemke’s algorithm for solving the linear complementarity problem.
4 citations
Cites background from "Linear complementarity, linear and ..."
...For recent books on the linear complementarity problem and its applications, see Cottle, Pang, and Stone [7], Murty [36], and Facchinei and Pang [15]....
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