Solution Of Linear Complementarity Problem Research Articles

Enhanced intersection cutting-plane approach for linear complementarity problems

In this paper, we develop an enhanced intersection cutting-plane algorithm for solving a mixed integer 0---1 bilinear programming formulation of the linear complementarity problem (LCP). The matrixM associated with the LCP is not assumed to possess any special structure, except that the corresponding feasible region is assumed to be bounded. A procedure is described to generate cuts that are deeper versions of the Tuy intersection cuts, based on a relaxation of the usual polar set. The proposed algorithm then attempts to find an LCP solution in the process of generating either a single or a pair of such strengthened intersection cuts. The process of generating these cuts involves a vertexranking scheme that either finds an LCP solution, or else these cuts eliminate the entire feasible region leading to the conclusion that no LCP solution exists. Computational experience on various test problems is provided.

Neural network approach to the solution of linear complementarity problems

AbstractThe paper presents the application of nonlinear neural optimization networks to solve the linear complementarity problem. Two different approaches are presented and investigated: one leading to linear and the second to quadratic optimization programming. The numerical results of illustrative examples are given and discussed.

Solution of linear complementarity problems using minimization with simple bounds

We define a minimization problem with simple bounds associated to the horizontal linear complementarity problem (HLCP). When the HLCP is solvable, its solutions are the global minimizers of the associated problem. When the HLCP is feasible, we are able to prove a number of properties of the stationary points of the associated problem. In many cases, the stationary points are solutions of the HLCP. The theoretical results allow us to conjecture that local methods for box constrained optimization applied to the associated problem are efficient tools for solving linear complementarity problems. Numerical experiments seem to confirm this conjecture.

On the local superlinear convergence of a Newton-type method for LCP under weak conditions

The paper proposes a damped Newton-type algorithm for linear complementarity problems (LCP). The algorithm is based on a special transformation of the LCP into an equivalent nonlinear system of equations and modifies a method proposed in a preceding paper of the author At first, results on global convergence are proved for LCP with Po-matrices. In particular, it is shown that all occuring subproblems are solvable and each accumulation point generated by the algorithm solves the LCP. Then, motivated by interior-point methods, the paper studies the local convergence to nondegenerated solutions of LCP with positive semi-definite matrices. To prove results, both on the local superlinear and the global convergence, suitably perturbed Newton subproblems will be introduced In contrast with many interior-point methods the algorithm can also have a superlinear rate of convergence if the LCP is degenerate. In particular, it converges Q-superlinearly under some strong second order condition. Furthermore, the Newton-...

A Linear Complementarity Approach to the Frictionless Gripper Problem

This article deals with static unilateral phenomena arising in grasping mechanisms and caused by the possible loss of contact between elastic grippers and rigid objects. The problem is formulated as a linear complementarity problem (LCP) that is derived from the equilibrium equations of forces and the compatibility equations for the displacements, as well as from the complementarity inequality restriction describing the contact. A variant of the infeasibility-reducing, complementary pivot al gorithm of Lemke is used for the numerical solution of the arising LCP. Form and force closure conditions are fulfilled in an auto matic way from the solution of the aforementioned LCP. The used algorithm guarantees rapid convergence to the solution of the problem. The theory is illustrated by numerical examples.

Parallel Implementation of Lemke's Algorithm on the Hypercube

A simple but very effective method for parallelizing Lemke's algorithm for the solution of linear complementarity problems is presented. Implementation details on a 32-node Intel iPSC/2 hypercube for problems of dimension up to 1000 are discussed. A speedup efficiency as high as 76% is achieved with 32 processing nodes for a problem with 500 variables and 250,000 nonzero elements. By combining the effects of concurrency and vectorization the computing time on the Intel iPSC/2 in some cases is reduced by a factor of 100. INFORMS Journal on Computing, ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.

On degeneracy in linear complementarity problems

Let M be an n× n matrix and q an nth order vector. Then the linear complementarity problem LCP( q, M) is defined as follows: determine x⩾ 0 such that w= Mx+ q⩾ 0 and x T w=0. A vector x which satisfies these conditions is called a solution of the problem, and a solution for which x i = w i =0 for at least one value of i is termed degenerate. If the solutions of LCP( q, M) are nondegenerate and their number is odd (even), we say that the solution set has odd (even) parity, and Murty has shown that this parity is determined uniquely by M. In this paper the idea of parity is extended to degenerate solutions and, through these, to solution sets containing both degenerate and nondegenerate solutions. These results are then used to give a generalization of Lemke's method and to analyse the stability of certain degenerate solutions of linear complementarity problems.

Geometric Properties of Hidden Minkowski Matrices

It is known that a vector satisfying a certain exponentially large set of inequalities related to a matrix M will allow the solution of linear complementarity problems with matrix M in n steps. It is shown that the set of such vectors, if nonempty, is the interior of a simplicial cone. The defining inequalities for this cone show that $M^T $ is hidden Minkowski if such a vector exists.

Acceptable solutions of linear complementarity problems

The theorem of Prager-Oettli for systems of linear equations is extended to linear complementarity problems. Moreover, using a result of Schaback, a roundoff-free formula for computer calculation of the obtained bound is given such that the strictness of the bound is guaranteed in spite of roundoff errors.

An enumerative method for the solution of linear complementarity problems

In this paper an enumerative method for the solution of the Linear Complementarity Problem (LCP) is presented. This algorithm either finds all the solutions of any general LCP (that is, no assumption is made concerning the class of the matrix), or it establishes that no such solution exists. The method is extended to the Second Linear Complementarity Problem (SLCP), A new problem which has been introduced for the solution of a general quadratic program.

An alternating direction implicit algorithm for the solution of linear complementarity problems arising from free boundary problems

We show that the alternating direction implicit algorithm of Peaceman—Rachford can be adapted to solve linear complementarity problems arising from free boundary problems. The alternating direction implicit algorithm is significantly faster than modified SOR algorithms. Some numerical results are given.

Multigrid Algorithms for the Solution of Linear Complementarity Problems Arising from Free Boundary Problems

Several free boundary problems (including saturated-unsaturated flow through porous dams, elastic-plastic torsion and cavitating journal bearings) can be formulated as linear complementarity problems of the following type: Find a nonnegative function u which satisfies prescribed boundary conditions on a given domain and which, furthermore, satisfies a linear elliptic equation at each point of the domain where u is greater than zero. We show that the multigrid FAS algorithm, which was developed by Brandt to solve boundary value problems for elliptic partial differential equations, can easily be adapted to handle linear complementarity problems. For large problems, the resulting algorithm, PFAS (projected full approximation scheme) is significantly faster than previous single-grid algorithms, since the computation time is proportional to the number of grid points on the finest grid. We then introduce two further multigrid algorithms, PFASMD and PFMG. PFASMD is a modification of PFAS which is considerably faster than PFAS. Using PFMG (projected full multigrid) it is possible to solve a linear complementarity problem to within truncation error using less work than the equivalent of seven Gauss–Seidel sweeps on the finest grid.

The efficient solution of linear complementarity problems for tridiagonal Minkowski matrices

The efficient solution of linear complementarity problems for tridiagonal Minkowski matrices

The solution of linear complementarity problems on an array processor

The Distributed Array Processor (DAP) manufactured by International Computers Limited is an array of 1-bit 200-nanosecond processors. The Pilot DAP on which the present work was done is a 32 X 32 array; the commercially available machine is a 64 X 64 array. We show how the projected SOR algorithm for the linear complementarity problem Aw ⩾ b, w ⩾ 0, w T ( Aw − b) = 0, can be adapted for use on the DAP when A is the finite-difference matrix corresponding to the difference approximation to the Laplace operator. Application is made to two linear complementarity problems arising, respectively, from two- and three-dimensional porous flow free boundary problems.

Solution of nonsymmetric linear complementarity problems by iterative methods

Convergence is established for iterative algorithms for the solution of the nonsymmetric linear complementarity problem of findingz such thatMz+q⩾0,z⩾0,zT(Mz+q)=0, whereM is a givenn×n real matrix, not necessarily symmeetric, andq is a givenn-vector. It is first shown that, if the spectral radius of a matrix related toM is less than one, then the iterates generated by the general algorithm converge to a solution of the linear complementarity problem. It turns out that convergence properties are quite similar to those of linear systems of equations. As specific cases, two important classes of matrices, Minkowski matrices and quasi-dominant diagonal matrices, are shown to satisfy this convergence condition.

A Least-Element Theory of Solving Linear Complementarity Problems as Linear Programs

In a previous report (Cottle, R. W., J. S. Pang. 1978. On solving linear complementary problems as linear programs. Math. Programming Stud. 7 88–107.), the authors have established a least-element interpretation to Mangasarian's theory (Mangasarian, O. L. 1976. Linear complementarity problems solvable by a single linear program. Math. Programming 10 263–270; Mangasarian, O. L. 1975. Solution of linear complementarity problems by linear programming. In G. W. Watson, ed. Numerical Analysis, Dundée 1975. Lecture Notes in Mathematics, No. 506, Springer-Verlag, Berlin, 166–175.) of formulating some linear complementarity problems as linear programs. In the present report, we extend our previous analysis to a more general class of linear complementarity problems investigated in Mangasarian (Mangasarian, O. L. 1978. Characterization of linear complementarity problems as linear programs. Math. Programming Stud. 7 74–87.), Our purposes are (1) to demonstrate how solutions to these problems can be generated from least elements of polyhedral sets and (2) to investigate how these “least-element solutions” are related to the solutions obtained by the linear programming approach as proposed by Mangasarian.

Solution of symmetric linear complementarity problems by iterative methods

A unified treatment is given for iterative algorithms for the solution of the symmetric linear complementarity problem: $$Mx + q \geqslant 0, x \geqslant 0, x^T (Mx + q) = 0$$ , whereM is a givenn×n symmetric real matrix andq is a givenn×1 vector. A general algorithm is proposed in which relaxation may be performed both before and after projection on the nonnegative orthant. The algorithm includes, as special cases, extensions of the Jacobi, Gauss-Seidel, and nonsymmetric and symmetric successive over-relaxation methods for solving the symmetric linear complementarity problem. It is shown first that any accumulation point of the iterates generated by the general algorithm solves the linear complementarity problem. It is then shown that a class of matrices, for which the existence of an accumulation point that solves the linear complementarity problem is guaranteed, includes symmetric copositive plus matrices which satisfy a qualification of the type: $$Mx + q > 0 for some x in R^n $$ . Also included are symmetric positive-semidefinite matrices satisfying this qualification, symmetric, strictly copositive matrices, and symmetric positive matrices. Furthermore, whenM is symmetric, copositive plus, and has nonzero principal subdeterminants, it is shown that the entire sequence of iterates converges to a solution of the linear complementarity problem.

Complementarity Theorems for Linear Programming

Previous article Next article Complementarity Theorems for Linear ProgrammingA. C. WilliamsA. C. Williamshttps://doi.org/10.1137/1012015PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] Frank Eugene Clark, Mathematical Notes: Remark on the Constraint Sets in Linear Programming, Amer. Math. Monthly, 68 (1961), 351–352 MR1531192 0109.38204 CrossrefISIGoogle Scholar[2] A. J. Goldman, Resolution and separation theorems for polyhedral convex setsLinear inequalities and related systems, Annals of Mathematics Studies, no. 38, Princeton University Press, Princeton, N. J., 1956, 41–51, in [4] MR0089113 0072.37505 Google Scholar[3] A. J. Goldman and , A. W. Tucker, Theory of linear programmingLinear inequalities and related systems, Annals of Mathematics Studies, no. 38, Princeton University Press, Princeton, N.J., 1956, 53–97, in [4] MR0101826 0072.37601 Google Scholar[4] H. W. Kuhn and , A. W. Tucker, Linear Inequalities and Related Systems, Princeton University Press, Princeton, 1956 0072.37502 Google Scholar[5] A. C. Williams, Boundedness relations for linear constraint sets, Linear Algebra and Appl., 3 (1970), 129–141 10.1016/0024-3795(70)90009-1 MR0266612 0201.22003 CrossrefGoogle Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Sparse solutions to an underdetermined system of linear equations via penalized Huber loss6 November 2020 | Optimization and Engineering, Vol. 22, No. 3 Cross Ref Dynamic Non-diagonal Regularization in Interior Point Methods for Linear and Convex Quadratic Programming26 February 2019 | Journal of Optimization Theory and Applications, Vol. 181, No. 3 Cross Ref Necessary and Sufficient Conditions for Noiseless Sparse Recovery via Convex Quadratic SplinesMustafa Ç Pinar12 February 2019 | SIAM Journal on Matrix Analysis and Applications, Vol. 40, No. 1AbstractPDF (416 KB)A labeling algorithm for the sensitivity ranges of the assignment problemApplied Mathematical Modelling, Vol. 35, No. 10 Cross Ref Bibliography15 August 2011 Cross Ref Sensitivity Analysis and Dynamic Programming15 February 2011 Cross Ref Bibliography Cross Ref Condition measures and properties of the central trajectory of a linear programMathematical Programming, Vol. 83, No. 1-3 Cross Ref New characterizations of ℓ1 solutions to overdetermined systems of linear equationsOperations Research Letters, Vol. 16, No. 3 Cross Ref Limiting behavior of weighted central paths in linear programmingMathematical Programming, Vol. 65, No. 1-3 Cross Ref Stability of linearly constrained convex quadratic programsJournal of Optimization Theory and Applications, Vol. 64, No. 1 Cross Ref Marginal values in mixed integer linear programmingMathematical Programming, Vol. 44, No. 1-3 Cross Ref A theory of linear inequality systemsLinear Algebra and its Applications, Vol. 106 Cross Ref Boundedness relations in linear semi-infinite programmingAdvances in Applied Mathematics, Vol. 8, No. 1 Cross Ref A Variable-Complexity Norm Maximization ProblemO. L. Mangasarian and T. -H. Shiau17 July 2006 | SIAM Journal on Algebraic Discrete Methods, Vol. 7, No. 3AbstractPDF (698 KB)Simple computable bounds for solutions of linear complementarity problems and linear programs26 February 2009 Cross Ref On polyhedral extension of some LP theoremsMathematical Programming, Vol. 30, No. 1 Cross Ref Polyhedral extensions of some theorems of linear programmingMathematical Programming, Vol. 24, No. 1 Cross Ref Optimal simplex tableau characterization of unique and bounded solutions of linear programsJournal of Optimization Theory and Applications, Vol. 35, No. 1 Cross Ref Projection and Restriction Methods in Geometric Programming and Related Problems Cross Ref Representation of Convex Sets Cross Ref Projection and restriction methods in geometric programming and related problemsJournal of Optimization Theory and Applications, Vol. 26, No. 1 Cross Ref The complementary unboundedness of dual feasible solution sets in convex programmingMathematical Programming, Vol. 12, No. 1 Cross Ref Theorems on the dimension of convex setsLinear Algebra and its Applications, Vol. 12, No. 1 Cross Ref On the primal and dual constraint sets in geometric programmingJournal of Mathematical Analysis and Applications, Vol. 32, No. 3 Cross Ref Volume 12, Issue 1| 1970SIAM Review History Submitted:23 December 1968Published online:18 July 2006 InformationCopyright © 1970 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1012015Article page range:pp. 135-137ISSN (print):0036-1445ISSN (online):1095-7200Publisher:Society for Industrial and Applied Mathematics