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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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21 May 2019
TL;DR: The approach is based on a block Bard-type algorithm that applies low-rank downdates to a Cholesky factorization of the system matrix at each pivoting step, which gives up to 3.5x speed-up versus recomputing the factorization based on the index set.
Abstract: Simulating stiff physical systems is a requirement for numerous computer graphics applications, such as VR training for heavy equipment operation. However, iterative linear solvers often perform poorly in such cases, and direct methods involving a factorization of the system matrix are typically preferred for accurate and stable simulations. This can have a detrimental impact on performance, since factorization of the system matrix is costly for complex simulations. In this paper, we present a method for efficiently solving linear systems of stiff physical systems involving contact, where the dynamics are modeled as a mixed linear complementarity problem (MLCP). Our approach is based on a block Bard-type algorithm that applies low-rank downdates to a Cholesky factorization of the system matrix at each pivoting step. Further performance improvements are realized by exploiting low bandwidth characteristics of the factorization. Our method gives up to 3.5x speed-up versus recomputing the factorization based on the index set. Various challenging scenarios are used to demonstrate the advantages of our approach.
5 citations
TL;DR: The notion of a Z-map due to Riddell to the set-valued case is generalized and some conditions of the feasible set being a sublattice are presented and the least element problems are discussed under some strict pseudomonotonicity conditions.
Abstract: The purpose of this paper is devoted to the least element problems of feasible sets for vector complementarity problems under certain conditions. We generalize the notion of a Z-map due to Riddell to the set-valued case. Some conditions of the feasible set being a sublattice are presented and the least element problems are discussed under some strict pseudomonotonicity conditions.
5 citations
Cites background from "Linear complementarity, linear and ..."
...Other works on the relationship between nonlinear programming problems and nonlinear complementarity problems can be found in [11, 12, 18]....
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Journal Article•
TL;DR: A linear transformation is introduced and positive definite matrices are characterized as the matrices with corresponding semidefinite linear complementarity problem having unique solutions.
Abstract: The class of positive definite and positive semidefinite matrices is one of the most frequently encountered matrix classes both in theory and practice. In statistics, these matrices appear mostly with symmetry. However, in complementarity problems generally symmetry in not necessarily an accompanying feature. Linear complementarity problems defined by positive semidefinite matrices have some interesting properties such as the solution sets are convex and can be processed by Lemke’s algorithm as well as Graves’ principal pivoting algorithm. It is known that the principal pivotal transforms (PPTs) (defined in the context of linear complementarity problem) of positive semidefinite matrices are all positive semidefinite. In this article, we introduce the concept of generalized PPTs and show that the generalized PPTs of a positive semidefinite matrix are also positive semidefinite. One of the important characterizations of P -matrices (that is, the matrices with all principle minors positive) is that the corresponding linear complementarity problems have unique solutions. In this article, we introduce a linear transformation and characterize positive definite matrices as the matrices with corresponding semidefinite linear complementarity problem having unique solutions. Furthermore, we present some simplification procedure in solving a particular type of semidefinite linear complementarity problems involving positive definite matrices.
5 citations
01 Jan 2005
TL;DR: In this paper, a survey of developments on the Linear Complementarity Problem (LCP) since 1978 is presented, the year in which the International School of Mathematics on Variational Inequalities and Complementary Problems took place at the ‘Ettore Majorana’ Centre of Scientific Culture in Erice, Sicily.
Abstract: We survey developments on the Linear Complementarity Problem (LCP) since 1978, the year in which the International School of Mathematics on Variational Inequalities and Complementarity Problems took place at the ‘Ettore Majorana’ Centre of Scientific Culture in Erice, Sicily. This report will touch on matrix classes and the existence of solutions, complexity, degeneracy resolution, algorithms, software products, applications and generalizations of the LCP.
5 citations
TL;DR: In this article, the unique solution of the linear complementarity problem (LCP) is further discussed using the absolute value equations, and some new results are obtained to guarantee the unique solutions of the LCP for any real vector.
Abstract: In this note, the unique solution of the linear complementarity problem (LCP) is further discussed. Using the absolute value equations, some new results are obtained to guarantee the unique solution of the LCP for any real vector.
5 citations
Cites background from "Linear complementarity, linear and ..."
...For more detailed descriptions, one can refer to Cottle and Dantzig (1968), Cottle, Pang, and Stone (1992), Murty (1988), Schäfer (2004) and the references therein....
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...Example 3.3 (Murty, 1988) Let where e ∈ ℝn denote the column vector whose elements are all 1....
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