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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: A new class of computations that is motivated by a model of line and edge detection in primary visual cortex, which is based on a dynamic analog model of computation, a model that is classically used to motivate gradient descent algorithms that seek extrema of energy functionals.

4 citations

Journal ArticleDOI
TL;DR: A direct algorithm for the solution to the affine two-sided obstacle problem with an M-matrix has the polynomial bounded computational complexity O(n3) and is more efficient than those in (Numer. Algebra Appl. 2006).
Abstract: A direct algorithm for the solution to the affine two-sided obstacle problem with an M-matrix is presented. The algorithm has the polynomial bounded computational complexity O(n3) and is more efficient than those in (Numer. Linear Algebra Appl. 2006; 13:543–551). Copyright © 2010 John Wiley & Sons, Ltd.

4 citations


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

  • ...in [2, 4]) is used to solve the linear complementarity problem with A being a Z -matrix, the algorithm terminates after at most n principal pivot steps, with either a solution of the linear complementarity problem or the conclusion that it has no solution, and the computational complexity of the algorithm is at most O(n3)....

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Dissertation
01 Jan 2004
TL;DR: It is shown that the proper active set may be determined in general by solving a certain nonparametric quadratic programming problem.
Abstract: We consider the convex parametric quadratic programming problem when the end of the parametric interval is caused by a multiplicity of possibilities (“ties”). In such cases, there is no clear way for the proper active set to be determined for the parametric analysis to continue. In this thesis, we show that the proper active set may be determined in general by solving a certain nonparametric quadratic programming problem. We simplify the parametric quadratic programming problem with a parameter both in the linear part of the objective function and in the right-hand side of the constraints to a quadratic programming without a parameter. We break the analysis into three parts. We first study the parametric quadratic programming problem with a parameter only in the linear part of the objective function, and then a parameter only in the right-hand side of the constraints. Each of these special cases is transformed into a quadratic programming problem having no parameters. A similar approach is then applied to the parametric quadratic programming problem having a parameter both in the linear part of the objective function and in the right-hand side of the constraints.

3 citations

Journal ArticleDOI
TL;DR: In this paper, an exterior point algorithm for positive definite linear complementarity problems (LCPs) is introduced, which exploits the ellipsoid method to find a starting point in the case of PDLCPs and to check for the problem feasibility in case of PSDLCPs.
Abstract: An exterior point algorithm for positive definite (PD) and positive semidefinite (PSD) linear complementarity problems (LCPs) is introduced. The algorithm exploits the ellipsoid method to find a starting point in the case of positive definite linear complementarity problems (PDLCPs) and to check for the problem feasibility in case of positive semidefinite linear complementarity problems (PSDLCPs). The algorithm starts from a point on the boundary on which the complementarity condition is satisfied and generates a sequence of points on that same boundary. These points converge to the solution. The algorithm is modified to speed up the convergence for some PDLCPs and PSDLCPs that arise in certain mechanical models. A numerical example and a practical example in robotics are solved to test the algorithm.

3 citations

Journal ArticleDOI
01 Dec 2004-Pamm
TL;DR: In this article, the necessary conditions for optimal non-smooth control of rigid body mechanical systems with multiple impacts are stated in accordance with Pontryagin's Minimum Principle, and the Weierstrass-Erdmann (WE) conditions and contemporary impact theory are established in order to assess the optimality of an impact.
Abstract: In this report the necessary conditions for optimal non-smooth control of rigid body mechanical systems with multiple impacts will be stated in accordance with Pontryagin’s Minimum Principle. Criteria based on the Weierstrass-Erdmann (WE) conditions and contemporary impact theory will be established in order to assess the optimality of an impact.The determination of post-impact state and costate values from the pre-impact values will be possible in some cases. Mechanical systems subject to inequality (or unilateral) constraints and impact effects have recently been the object of significant interest in the mechanical engineering and applied mathematics scientific communities, in parallel the analysis and control of hybrid dynamical systems is an active area of investigation in the systems and control community. From the point of view of calculus of variations on which optimal control builds up on, this requires to consider the variations of piecewisesmooth maximizing trajectories. Mechanically, these non-smooth trajectories are obtained with impacts. If the allowable set of variations is extended to include piecewise-smooth trajectories, it is evident that a reduction of the cost functional can further be achieved in comparison to the case where only smooth optimal trajectories are considered, which constitute a subset of the piecewise-smooth maximizing trajectories. Indeed, there are even cases where a solution among the smooth extremizing trajectories does not exist because of reachability space considerations. In order to discuss impacts for a general rigid body system with Newton impact law and Coulomb friction, a linear complementarity problem (LCP) can be formulated in terms of impulses and contact kinematics variables that will enable the determination of the post-impact state. The solution of the LCP can be evaluated together with the Weierstrass-Erdmann condition so that the optimality of the impact can be assessed. The equation of motion of a general scleronomic mechanical system is given by: M(q)˜ i h(q; _

3 citations


Additional excerpts

  • ...λ̇ T = −∂H ∂q̇ = − ∂g ∂q̇ − λ M−1(q) q̇, τ ) ∂q̇ − p , ṗ = −∂H ∂q (9)...

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