Linear complementarity, linear and nonlinear programming
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Cites background from "Linear complementarity, linear and ..."
...Trouble also occurred with the MJN algorithm for the Murty matrices, where it could not even find a stationary point of the merit function....
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...In our first set of test problems, B is always the identity matrix and A ∈ Rn×n is an asymmetric matrix from various classes: Lotkin (a modification in the first row of a Hilbert matrix altered to all ones), Murty [23], Tridiagonal, S ×D or Q×D, where S, Q and D are symmetric, orthogonal and diagonal matrices, respectively, or A is randomly generated such that each element is uniformly distributed in the interval [−1, 1]....
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...In our first set of test problems, B is always the identity matrix and A ∈ R is an asymmetric matrix from various classes: Lotkin (a modification in the first row of a Hilbert matrix altered to all ones), Murty [23], Tridiagonal, S ×D or Q×D, where S, Q and D are symmetric, orthogonal and diagonal matrices, respectively, or A is randomly generated such that each element is uniformly distributed in the interval [−1, 1]....
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23 citations
Cites background or methods from "Linear complementarity, linear and ..."
...For more details on the linear complementarity problem, we refer to Cottle, Pang and Stone (1992) and Murty (1988). As mentioned in the introduction, we are interested in addressing the following problem: Under what conditions on matrix M does the problem LCP(q , M) have an integer solution for every integer q for which it has a solution? To this effect, we study some of the wellknown classes of matrices in linear complementarity and arrive at some characterizations. We follow the notations and terminology as given in Cottle, Pang, and Stone (1992) and Murty (1988). We consider both matrices and vectors having integer entries only, unless otherwise specified explicitly. Let M √ Rn1n . For subsets J , K ⊆ {1, . . . , n}, we denote by MJK the submatrix of M , with rows and columns corresponding to the index sets J and K , respectively. The matrix MJJ for J ⊆ {1, . . . , n} denotes a principal submatrix of M , when ÉJÉ Å k , MJJ is called the principal submatrix of order k . Then, the determinant of MJJ , is called a principal minor of order k . For j√ {1, . . . , n}, the j th column of M is denoted by M.j . We represent the (n 0 1) 1 (n 0 1) principal submatrix of M obtained by deleting the i th row and the i th column by Mii . For any J ⊆ {1, . . . , n}, JV denotes the set {1, 2, . . . , n} "J . By eJ , we denote the vector of 1’s, of size ÉJÉ. By ej , we denote the unit vector with 1 in the j th coordinate and the rest of its entries zero. For a J ⊆ {1, . . . , n}, we denote by C(J) the complementary matrix whose columns are C(J).j Å 0M.j whenever j √ J and C(J).jÅ I.j otherwise. Then pos C(J)Å: {C(J)x : x√ is called a complementary n R } / cone relative to the matrix M . A complementary cone pos C(J) is said to be full , if det MJJ x 0. We denote by K(M) , the set of vectors q √ R for which (1) has a solution. Clearly, q √ K(M) if and only if there exists a complementary cone pos C(J) , such that q √ pos C(J) . For further details on complementary cones, we refer to Cottle, Pang and Stone (1992) and Murty (1988)....
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...For more details on the linear complementarity problem, we refer to Cottle, Pang and Stone (1992) and Murty (1988)....
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...…N (all proper principal minors negative and determinant positive) matrices (see Cottle, Pang, and Stone 1992; Mohan, Parthasarathy, and Sridhar 1994; Murty 1988; Olech, Parathasarathy, and Ravindran 1989; and Olech, Parathasarathy, and Ravindran 1991) (the last two are known as E1-matrices) ....
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...For more details on the linear complementarity problem, we refer to Cottle, Pang and Stone (1992) and Murty (1988). As mentioned in the introduction, we are interested in addressing the following problem: Under what conditions on matrix M does the problem LCP(q , M) have an integer solution for every integer q for which it has a solution? To this effect, we study some of the wellknown classes of matrices in linear complementarity and arrive at some characterizations....
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...We follow the notations and terminology as given in Cottle, Pang, and Stone (1992) and Murty (1988)....
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