scispace - formally typeset
Search or ask a question
Book

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.
Citations
More filters
Journal ArticleDOI
TL;DR: In this article, Tucker's, Cottle's and Dantzig's principal pivoting methods are specialized as diagonal and exchange pivots for the linear complementarity problem obtained from a convex quadratic program.
Abstract: Three generalizations of the criss-cross method for quadratic programming are presented here. Tucker’s, Cottle’s and Dantzig’s principal pivoting methods are specialized as diagonal and exchange pivots for the linear complementarity problem obtained from a convex quadratic program A finite criss-cross method, based on least-index resolution, is constructed for solving the LCP. In proving finiteness, orthogonality properties of pivot tableaus and positive semidefiniteness of quadratic matrices are used In the last section some special cases and two further variants of the quadratic criss-cross method are discussed. If the matrix of the LCP has full rank, then a surprisingly simple algorithm follows, which coincides with Murty’s ‘Bard type schema’ in the P matrix case

24 citations

20 Nov 2003
TL;DR: A splitting method for solving LCP based models of dry frictional contact problems in rigid multibody systems based on box MLCP solver is presented and their performance is compared both on random problems and on simulation data.
Abstract: A splitting method for solving LCP based models of dry frictional contact problems in rigid multibody systems based on box MLCP solver is presented. Since such methods rely on fast and robust box MLCP solvers, several methods are reviewed and their performance is compared both on random problems and on simulation data. We provide data illustrating the convergence rate of the splitting method which demonstrates that they present a viable alternative to currently available methods. CR Categories: G.1.6 [Mathematics of Computing]: Optimization—Nonlinear Programming I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling—Physically based modeling I.3.7 [Computer Graphics]: Three-Dimensional Graphics and Realism—Virtual Reality I.6.8 [Simulation and Modeling]: Types of Simulation—Animation

24 citations

Journal ArticleDOI
TL;DR: This book provides a numerical foundation for linear complementarity problems, especially suited for use in computer graphics, and provides pseudo code for all the numerical methods, which should be comprehensible by any computer scientist with rudimentary programming skills.
Abstract: Linear complementarity problems (LCPs) have for many years been used in physics-based animation to model contact forces between rigid bodies in contact. More recently, LCPs have found their way into the realm of fluid dynamics. Here, LCPs are used to model boundary conditions with fluid-wall contacts. LCPs have also started to appear in deformable models and granular simulations. ere is an increasing need for numerical methods to solve the resulting LCPs with all these new applications.is book provides a numerical foundation for suchmethods, especially suited for use in computer graphics. is book is mainly intended for a researcher/Ph.D. student/post-doc/professor who wants to study the algorithms and do more work/research in this area. Programmers might have to invest some time brushing up on math skills, for this we refer to Appendices A and B. e reader should be familiar with linear algebra and differential calculus. We provide pseudo code for all the numerical methods, which should be comprehensible by any computer scientist with rudimentary programming skills. e reader can find an online supplementary code repository, containing Matlab implementations of many of the core methods covered in these notes, as well as a few Python implementations [Erleben, 2011].

24 citations


Cites background or methods from "Linear complementarity, linear and ..."

  • ...…reformulating the LCP into other types of problems. e first reformulation we present is known as the minimum map reformulation [Cottle et al., 1992, Murty, 1988, Pang, 1990] h.x; y/ min.x; y/ (1.17) An alternative convenient notation is min.x; y/ h.x; y/, which we use later whenwe generalize…...

    [...]

  • ...Another popular reformulation is based on the Fischer-Burmeister function, which is defined as [Cottle et al., 1992, Fischer, 1992, Murty, 1988] FB.x; y/ p x2 C y2 x y (1.21) 1.1....

    [...]

Journal ArticleDOI
TL;DR: A number of properties of Lemke paths are investigated, motivated by the d -step conjecture for linear programming, including families of LemK paths, for which the length of the shortest grows exponentially, joining pairs of vertices of a sequence of polytopes.
Abstract: Lemke paths are often used in the solution of nonlinear programming problems. We investigate a number of properties of Lemke paths, motivated by the d-step conjecture for linear programming. Some negative results are presented, including families of Lemke paths, for which the length of the shortest grows exponentially, joining pairs of vertices of a sequence of polytopes.

24 citations

Journal ArticleDOI
TL;DR: In this paper, the authors consider variational inequality problems where the convex set under consideration is a bounded polytope and define an associated box constrained minimization problem and prove that, under a general condition on the Jacobian, the stationary points of the minimization problems are solutions of variational inequalities.
Abstract: We consider variational inequality problems where the convex set under consideration is a bounded polytope. We define an associated box constrained minimization problem and we prove that, under a general condition on the Jacobian, the stationary points of the minimization problems are solutions of the variational inequality problem. The condtion includes the case where the operator is monotone. Based on this result we develop an algorithm that can solve large scale problems. We present numerical experiments.

24 citations


Cites background from "Linear complementarity, linear and ..."

  • ...In recent papers, we considered the reduction of convex linearly constrained minimization problems to box constrained minimization [11] and, more recently, we studied linear complementarity problems ([19], [5]) under the same framework [12]....

    [...]