Linear complementarity, linear and nonlinear programming

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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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Proceedings Article•DOI•

Optimization-based animation

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Victor Milenkovic¹, Harald Schmidl¹•Institutions (1)

University of Miami¹

01 Aug 2001

TL;DR: An efficient optimization-based animation (OBA) algorithm is presented which can simulate scenes with many convex three-dimensional bodies settling into stacks and other “crowded” arrangements and no other packages that can simulate similarly complex scenes in a practical amount of time are found.

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Abstract: Current techniques for rigid body simulation run slowly on scenes with many bodies in close proximity. Each time two bodies collide or make or break a static contact, the simulator must interrupt the numerical integration of velocities and accelerations. Even for simple scenes, the number of discontinuities per frame time can rise to the millions. An efficient optimization-based animation (OBA) algorithm is presented which can simulate scenes with many convex three-dimensional bodies settling into stacks and other “crowded” arrangements. This algorithm simulates Newtonian (second order) physics and Coulomb friction, and it uses quadratic programming (QP) to calculate new positions, momenta and accelerations strictly at frame times. Contact points are synchronized at the end of each frame. The extremely small integration steps inherent to traditional simulation techniques are avoided. Non-convex bodies are simulated as unions of convex bodies. Links and joints are simulated successfully with bi-directional constraints. A hybrid of OBA and retroactive detection (RD) has been implemented as well. A review of existing work finds no other packages that can simulate similarly complex scenes in a practical amount of time.

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88 citations

Journal Article•DOI•

Iterative Solution of Piecewise Linear Systems

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Luigi Brugnano¹, Vincenzo Casulli²•Institutions (2)

University of Florence¹, University of Trento²

01 Nov 2007-SIAM Journal on Scientific Computing

TL;DR: In the present paper a simple Newton-type procedure for certain piecewise linear systems is derived and shown to have a finite termination property, i.e., it converges to the exact solution in a finite number of steps.

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Abstract: The correct formulation of numerical models for free-surface hydrodynamics often requires the solution of special linear systems whose coefficient matrix is a piecewise constant function of the solution itself. In so doing one may prevent the development of unrealistic negative water depths. The resulting piecewise linear systems are equivalent to particular linear complementarity problems whose solutions could be obtained by using, for example, interior point methods. These methods may have a favorable convergence property, but they are purely iterative and convergence to the exact solution is proven only in the limit of an infinite number of iterations. In the present paper a simple Newton-type procedure for certain piecewise linear systems is derived and discussed. This procedure is shown to have a finite termination property, i.e., it converges to the exact solution in a finite number of steps, and, actually, it converges very quickly, as confirmed by a few numerical tests.

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88 citations

Journal Article•DOI•

Operator-Splitting Methods for Monotone Affine Variational Inequalities, with a Parallel Application to Optimal Control

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Jonathan Eckstein¹, Michael C. Ferris²•Institutions (2)

Rutgers University¹, University of Wisconsin-Madison²

01 Feb 1998-Informs Journal on Computing

TL;DR: This article applies matrix-splitting-like method to discrete-time optimal control problems formulated as extended linear-quadratic programs in the manner advocated by Rockafellar and Wets, and develops a highly parallel algorithm.

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Abstract: This article applies splitting techniques developed for set-valued maximal monotone operators to monotone affine variational inequalities, including, as a special case, the classical linear complementarity problem. We give a unified presentation of several splitting algorithms for monotone operators, and then apply these results to obtain two classes of algorithms for affine variational inequalities. The second class resembles classical matrix splitting, but has a novel "under-relaxation" step, and converges under more general conditions. In particular, the convergence proofs do not require the affine operator to be symmetric. We specialize our matrix-splitting-like method to discrete-time optimal control problems formulated as extended linear-quadratic programs in the manner advocated by Rockafellar and Wets. The result is a highly parallel algorithm, which we implement and test on the Connection Machine CM-5 computer family.

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88 citations

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

...It is well known that when ` 0 and u 1, theproblem reduces to the linear complementarity problem (LCP) [5, 6, 24] of nding some x 2 <nsatisfying x 0 Mx + q 0 hx;Mx+ qi = 0 : (3)This paper is restricted to the monotone case of avi(M; q;B), where M is positive semide nite,although not necessarily symmetric....
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...We further note that by letting k > 0 in Proposition 6, the iteration (26) will still convergeeven if (I + cM) 1(rB(zk) cq) is computed inexactly, provided the accuracy improves su cientlywith k.Consider brie y the special case of the LCP (3), where B = n+....
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...We show that they take a very simple form indeed in the caseof the LCP (3); see (27) below....
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...The feasible region of avi(M; q;B) is de ned asfeas(M; q;B) := fx jMx+ q 2 (recB) ; x 2 Bg ;where recC denotes the recession cone of a set C de ned byrecC := fd 2 <n j x+ cd 2 C 8 x 2 C; c 0g ;and \ " denotes the dual cone operation de ned byK := fy j hy; vi 0; 8 v 2 K g :By way of illustration, in the special case of the LCP (3) the feasible set is fx jMx+ q 0; x 0g,while the solution set consists of all elements of the feasible set that also satisfy the complementaritycondition hx;Mx+ qi = 0.2.1 Simple Splitting SchemesWe now introduce two operators that constitute splitting of avi(M; q;B)....
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...It is well known that when ` 0 and u 1, the problem reduces to the linear complementarity problem (LCP) [5, 6, 24] of nding some x 2 <n satisfying x 0 Mx + q 0 hx;Mx+ qi = 0 : (3) This paper is restricted to the monotone case of avi(M; q;B), where M is positive semide nite, although not necessarily symmetric....
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Journal Article•DOI•

The P-matrix problem is co-NP-complete

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Gregory E. Coxson¹•Institutions (1)

University of Wisconsin-Madison¹

15 Apr 1994-Mathematical Programming

TL;DR: It is shown that given any matrix family belonging to the class of matrix polytopes with hypercube domains and rank-one perturbation matrices, a class which contains the intervalMatrices, testing singularity reduces to testing whether a certain matrix is not a P-matrix.

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Abstract: Recently Rohn and Poljak proved that for interval matrices with rank-one radius matrices testing singularity is NP-complete. This paper will show that given any matrix family belonging to the class of matrix polytopes with hypercube domains and rank-one perturbation matrices, a class which contains the interval matrices, testing singularity reduces to testing whether a certain matrix is not a P-matrix. It follows from this result that the problem of testing whether a given matrix is a P-matrix is co-NP-complete.

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88 citations

Journal Article•DOI•

The complementarity class of hybrid dynamical systems

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Wpmh Maurice Heemels¹, Bernard Brogliato²•Institutions (2)

Eindhoven University of Technology¹, French Institute for Research in Computer Science and Automation²

01 Jan 2003-European Journal of Control

TL;DR: In this paper, a summary of the main applications and properties of dynamical complementarity systems together with connections to other hybrid model classes is provided, and the main mathematical tools which allow one to study complementarity system are presented briefly.

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87 citations

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Linear complementarity, linear and nonlinear programming

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