The generalized linear complementarity problem revisited

doi:10.1007/BF02592211

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The generalized linear complementarity problem revisited

Journal Article•DOI•

The generalized linear complementarity problem revisited

S. R. Mohan¹, S. K. Neogy¹, R. Sridhar²•Institutions (2)

Indian Statistical Institute¹, Indira Gandhi Institute of Development Research²

31 Aug 1996-Mathematical Programming (Springer-Verlag New York, Inc.)-Vol. 74, Iss: 2, pp 197-218

TL;DR: This paper forms this problem as a linear complementarity problem with a square matrixM, a formulation which is different from a similar formulation given earlier by Lemke, and shows that the class of vertical block matrices which Cottle and Dantzig's algorithm can process is the same as theclass of equivalent square matrices in Lemke's algorithm.

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Abstract: Given a vertical block matrixA, we consider in this paper the generalized linear complementarity problem VLCP(q, A) introduced by Cottle and Dantzig. We formulate this problem as a linear complementarity problem with a square matrixM, a formulation which is different from a similar formulation given earlier by Lemke. Our formulation helps in extending many well-known results in linear complementarity to the generalized linear complementarity problem. We also show that the class of vertical block matrices which Cottle and Dantzig's algorithm can process is the same as the class of equivalent square matrices which Lemke's algorithm can process. We also present some degree-theoretic results on a vertical block matrix.

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Journal Article•DOI•

Convex piecewise-linear fitting

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Alessandro Magnani¹, Stephen Boyd¹•Institutions (1)

Stanford University¹

01 Mar 2009-Optimization and Engineering

TL;DR: The method described, which is a variation on the K-means algorithm for clustering, seems to work well in practice, at least on data that can be fit well by a convex function.

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Abstract: We consider the problem of fitting a convex piecewise-linear function, with some specified form, to given multi-dimensional data. Except for a few special cases, this problem is hard to solve exactly, so we focus on heuristic methods that find locally optimal fits. The method we describe, which is a variation on the K-means algorithm for clustering, seems to work well in practice, at least on data that can be fit well by a convex function. We focus on the simplest function form, a maximum of a fixed number of affine functions, and then show how the methods extend to a more general form.

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220 citations

Cites background from "The generalized linear complementar..."

...…al. 1990), extraction of straight lines in aerial images (Venkateswar and Chellappa 1992), global optimization (Mangasarian et al. 2005), compression of chemical process data (Bakshi and Stephanopoulos 1996), and circuit modeling (Julian et al. 1998; Chua and Deng 1986; Vandenberghe et al. 1989)....
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...2005), compression of chemical process data (Bakshi and Stephanopoulos 1996), and circuit modeling (Julian et al. 1998; Chua and Deng 1986; Vandenberghe et al. 1989)....
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Journal Article•DOI•

A non-interior continuation method for generalized linear complementarity problems

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Jiming Peng¹, Zhenghua Lin²•Institutions (2)

Delft University of Technology¹, Jilin University²

01 Dec 1999-Mathematical Programming

TL;DR: A non-interior continuation method for solving generalized linear complementarity problems (GLCP) introduced by Cottle and Dantzig is proposed, based on a smoothing function derived from the exponential penalty function first introduced by Kort and Bertsekas for constrained minimization.

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Abstract: In this paper, we propose a non-interior continuation method for solving generalized linear complementarity problems (GLCP) introduced by Cottle and Dantzig. The method is based on a smoothing function derived from the exponential penalty function first introduced by Kort and Bertsekas for constrained minimization. This smoothing function can also be viewed as a natural extension of Chen-Mangasarian’s neural network smooth function. By using the smoothing function, we approximate GLCP as a family of parameterized smooth equations. An algorithm is presented to follow the smoothing path. Under suitable assumptions, it is shown that the algorithm is globally convergent and local Q-quadratically convergent. Few preliminary numerical results are also reported.

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71 citations

Cites background or methods or result from "The generalized linear complementar..."

...3 in [15], and [21,32,36]) and then apply the smoothing methods for CPs to solve the reformulated problem....
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...[32] A square submatrix of N of ordern is called a representative submatrix, if its i th row is drawn from thei th block Ni of N, for i = 1, ....
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...Several authors have studied this problem and some numerical methods have been proposed to solve GLCP, the interested readers are referred to [12,13,31,32,39,42] and the references therein....
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...[32] A vertical block matrixN of type(m1, ....
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...Since a GLCP with a vertical block P-matrix is of typeR0 and its solution is also unique [32], we get the following result as a direct consequence of the above theorem....
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Journal Article•DOI•

A Smoothing Newton Method for Extended Vertical Linear Complementarity Problems

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Houduo Qi, Li-Zhi Liao

01 Oct 1999-SIAM Journal on Matrix Analysis and Applications

TL;DR: It is proved that every accumulation point of this sequence is a solution of EVLCP(M, q) under the assumption of row ${\cal W}_0$-property, and if row W-property holds at the solution point, then the convergence rate is quadratic.

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Abstract: In this paper, we reformulate the extended vertical linear complementarity problem (EVLCP(m,q)) as a nonsmooth equation H(t,x)=0, where $H: \mbox{\smallBbb R}^{n+1} \to \mbox{\smallBbb R}^{n+1}$, $t \in \mbox{\smallBbb R}$ is a parameter variable, and $x \in \mbox{\smallBbb R}$ is the original variable. H is continuously differentiable except at such points (t,x) with t=0. Furthermore H is strongly semismooth. The reformulation of EVLCP(m, q) as a nonsmooth equation is based on the so-called aggregation (smoothing) function. As a result, a Newton-type method is proposed which generates a sequence {wk=(tk,xk)} with all tk >0. We prove that every accumulation point of this sequence is a solution of EVLCP(M, q) under the assumption of row ${\cal W}_0$-property. If row ${\cal W}$-property holds at the solution point, then the convergence rate is quadratic. Promising numerical results are also presented.

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61 citations

Cites background from "The generalized linear complementar..."

...Several papers address both VLCP and EVLCP; see [15, 14, 29, 21, 16, 19] and the references therein....
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...For example Mohan, Neogy, and Sridhar [21] reformulate VLCP as an LCP and extend many well-known results in LCP to VLCP....
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Journal Article•DOI•

Exceptional Families of Elements for Continuous Functions: Some Applications to Complementarity Theory

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George Isac¹, A. Carbone•Institutions (1)

Royal Military College of Canada¹

01 Sep 1999-Journal of Global Optimization

TL;DR: Using the topological degree and the concept of exceptional family of elements for a continuous function, this work proves a very general existence theorem for the nonlinear complementarity problem.

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Abstract: Using the topological degree and the concept of exceptional family of elements for a continuous function, we prove a very general existence theorem for the nonlinear complementarity problem. This result is an alternative theorem. A generalization of Karamardian‘s condition and the asymptotic monotonicity are also introduced. Several applications of the main results are presented.

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39 citations

Journal Article•DOI•

On the equivalence of linear complementarity problems

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B. De Schutter¹, Wpmh Maurice Heemels², Alberto Bemporad•Institutions (2)

Delft University of Technology¹, Eindhoven University of Technology²

01 Aug 2002-Operations Research Letters

TL;DR: It is shown that the Extended Linear Complementarity Problem (ELCP) can be recast as a standard Linear Complementation Problem (LCP) provided that the surplus variables or the feasible set of the ELCP are bounded.

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27 citations

Cites background from "The generalized linear complementar..."

...• In [32, 35] it has been shown that the Vertical LCP can be reformulated as an LCP....
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References

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Book•

The Linear Complementarity Problem

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Richard W. Cottle, Jong-Shi Pang, Richard Stone

18 Feb 1992

TL;DR: In this article, the authors present an overview of existing and multiplicity of degree theory and propose pivoting methods and iterative methods for degree analysis, including sensitivity and stability analysis.

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Abstract: Introduction. Background. Existence and Multiplicity. Pivoting Methods. Iterative Methods. Geometry and Degree Theory. Sensitivity and Stability Analysis. Chapter Notes and References. Bibliography. Index.

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2,897 citations

Journal Article•DOI•

Equilibrium Points of Bimatrix Games

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C. E. Lemke, J. T. Howson

01 Jun 1964-Journal of The Society for Industrial and Applied Mathematics

TL;DR: An algebraic proof of the existence of equilibrium points for two-person non-zero-sum games is given in this paper, leading to an efficient scheme for computing an equilibrium point, which is valid for any ordered field.

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Abstract: An algebraic proof is given of the existence of equilibrium points for bimatrix (or two-person, non-zero-sum) games. The proof is constructive, leading to an efficient scheme for computing an equilibrium point. In a nondegenerate case, the number of equilibrium points is finite and odd. The proof is valid for any ordered field.

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1,087 citations

"The generalized linear complementar..." refers methods in this paper

...The algorithm presented by Lemke and Howson [ 19 ] to compute an equilibrium pair of strategies to a bimatrix game, later extended by Lemke [18] to solve a LCP(q, M) contributed significantly to the development of the linear complementarity theory....
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...However we can apply the algorithm of Lemke and Howson [ 19 ] to this problem and compute an equilibrium point....
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Book•

Linear complementarity, linear and nonlinear programming

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Katta G. Murty

01 Jan 1988

1,012 citations

Journal Article•DOI•

Bimatrix Equilibrium Points and Mathematical Programming

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C. E. Lemke

01 May 1965-Management Science

TL;DR: In this paper, simple constructive proofs are given of solutions to the matric matric system Mz − ω = q; z ≧ 0; ω ≧ 1; zT = 0, for various kinds of data M, q, which embrace quadratic programming and the problem of finding equilibrium points of bimatrix games.

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Abstract: Some simple constructive proofs are given of solutions to the matric system Mz − ω = q; z ≧ 0; ω ≧ 0; and zT ω = 0, for various kinds of data M, q, which embrace the quadratic programming problem and the problem of finding equilibrium points of bimatrix games. The general scheme is, assuming non-degeneracy, to generate an adjacent extreme point path leading to a solution. The scheme does not require that some functional be reduced.

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966 citations

"The generalized linear complementar..." refers methods in this paper

...The algorithm presented by Lemke and Howson [19] to compute an equilibrium pair of strategies to a bimatrix game, later extended by Lemke [ 18 ] to solve a LCP(q, M) contributed significantly to the development of the linear complementarity theory....
[...]

Journal Article•

Equilibrium points of bimatrix games

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C. E. Lemke

01 Jan 1962-Siam Journal on Applied Mathematics

TL;DR: An algebraic proof of the existence of equilibrium points for two-person non-zero-sum games is given in this article, leading to an efficient scheme for computing an equilibrium point, which is valid for any ordered field.

...read moreread less

785 citations