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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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Journal ArticleDOI
TL;DR: In this article, the problem of tracing the equilibrium path in large displacement frictionless contact problems is solved by writing the problem as a system of non-linear B-differentiable functions, where the non-differentiability due to the presence of unilateral contact constraints is overcome.
Abstract: This work is concerned with the problem of tracing the equilibrium path in large displacement frictionless contact problems. Conditions for the detection of critical points along the equilibrium path are also given. By writing the problem as a system of non-linear B-differentiable functions, the non-differentiability due to the presence of the unilateral contact constraints is overcome. The path-following algorithm is given as a predictor-corrector method, where the corrector part is performed using Newton's method for B-differentiable functions. A new type of displacement constraints are introduced where the constraining displacement node may change during the corrector iterations. Furthermore it is shown that, in addition to the usual bifurcation and limit points, bifurcation is possible or the equilibrium path may have reached an end point even if the stiffness matrix is non-singular.

24 citations

Journal ArticleDOI
TL;DR: In this article, the estimation of the first order conditions of a traditional transportation model is discussed, which is typically present in so-called Takayama-Judge type spatial price equilibrium models, and the proposed method uses bi-level programming techniques to minimize a weighted least squares criterion under the restriction that the estimated parameters satisfy the Kuhn-Tucker conditions.

24 citations

Journal ArticleDOI
TL;DR: In this article, a physics-based model that follows the principle of closed-form forward kinematics and constraint-based dynamics to present the dual-crane mechanism mathematically is presented.

24 citations

Journal Article
TL;DR: In this article, a hybrid algorithm combining a projection technique and a modified Josephy-Newton method is proposed to solve the asymmetric eigenvalue comple- mentarity problem by finding a stationary point of the gap function and the regularized gap function.
Abstract: In this paper, the solution of the asymmetric eigenvalue comple- mentarity problem (EiCP) is investigated by means of a variational inequality formulation. This problem is then solved by finding a stationary point of the gap function and the regularized gap function. A nonlinear programming formulation of the EiCP results from the gap function. A hybrid algorithm combining a projection technique and a modified Josephy-Newton method is proposed to solve the EiCP by finding a stationary point of the regularized gap function. Numerical results show that the proposed method can in gen- eral solve EiCPs efficiently.

23 citations


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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Journal ArticleDOI
TL;DR: A peeling algorithm for finding an integer solution for linear complementarity problems is given and related results are derived.
Abstract: We consider the problem of finding an integer solution to a linear complementarity problem. We introduce the class I of matrices for which the corresponding linear complementarity problem has an integer complementary solution for every vector, q, for which it has a solution. Strong principal unimodularity forms a sufficient condition for inclusion in the class I. It is also shown to be necessary for some well-known classes of matrices, including the class of positive semidefinite matrices. This is used in deriving necessary and sufficient conditions for a convex quadratic program to have an integer optimal solution for all b and c for which it has an optimal solution. Characterizations are also derived for some other well-known classes of matrices in linear complementarity to belong to the class I. In the end, a peeling algorithm for finding integer solution for linear complementarity problems is given and related results are derived.

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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