Solution Of Linear Complementarity Problem Research Articles

A two-level domain decomposition algorithm for linear complementarity problem

In this paper, a two-level domain decomposition algorithm for linear complementarity problem (LCP) is introduced. Inner and outer approximation sequences to the solution of LCP are generated by the proposed algorithm. The algorithm is proved to be convergent and can reach the solution of the problem within finite steps. Some simple numerical results are presented to show the effectiveness of the proposed algorithm.

The s-monotone index selection rule for criss-cross algorithms of linear complementarity problems

Abstract In this paper we introduce the s-monotone index selection rules for the well-known criss-cross method for solving the linear complementarity problem (LCP). Most LCP solution methods require a priori information about the properties of the input matrix. One of the most general matrix properties often required for finiteness of the pivot algorithms (or polynomial complexity of interior point algorithms) is sufficiency. However, there is no known polynomial time method for checking the sufficiency of a matrix (classification of column sufficiency of a matrix is co-NP-complete). Following the ideas of Fukuda, Namiki and Tamura, using Existentially Polynomial (EP)-type theorems, a simple extension of the crisscross algorithm is introduced for LCPs with general matrices. Computational results obtained using the extended version of the criss-cross algorithm for bi-matrix games and for the Arrow-Debreu market equilibrium problem with different market size is presented.

Investigation of rigid-plastic beams subjected to impact using linear complementarity

As a means for determining the response of a rigid-plastic structure to impact, formulation of the problem as a Linear Complementarity Problem (LCP) through finite element modelling is shown to offer the benefits of a systematic numerical process which is also automatic from start-up to standstill. Solution of a recurrent sequence of LCPs traces the development of plastic deformation discretely through time, as the structure responds to impact. It turns out that Lemke’s algorithm, chosen as a robust LCP solver, requires extra support to avoid cycling, very prevalent in this type of problem.Investigation of a central transverse impact on a simply supported beam with infinite resistance to shear deformation provided the opportunity for an LCP solution to be compared with a continuum representation of the problem. With simple plastic hinges, deforming only in bending, the improvement in accuracy that normally follows refinement of a finite element mesh was partly undermined by the effects of numerical difficulty with the modelling. A second investigation was made into the central transverse impact of a beam, this time with fixed supports, but more especially one in which the response involved both shear and bending deformations, interacting through the normal flow rule of a simple rectangular yield criterion. The numerical difficulty of the previous investigation completely disappeared and comparisons with available closed-form solutions confirmed the accuracy and convergence of the LCP solution.

A continuation method for linear complementarity problems with P 0 matrix

In this article, we propose a new continuation method for solving the linear complementarity problem (LCP). The method solves one system of linear equations and carries out only a one-line search at each iteration. The continuation method is based on a modified smoothing function. The existence and continuity of a smooth path for solving the LCP with a P 0 matrix are discussed. We investigate the boundedness of the iteration sequence generated by our continuation method under the assumption that the solution set of the LCP is nonempty and bounded. It is shown to converge to an LCP solution globally linearly and locally superlinearly without the assumption of strict complementarity at the solution under suitable assumption. In addition, some numerical results are also reported in this article.

Nondegenerate Piecewise Linear Systems: A Finite Newton Algorithm and Applications in Machine Learning

We investigate Newton-type optimization methods for solving piecewise linear systems (PLSs) with nondegenerate coefficient matrix. Such systems arise, for example, from the numerical solution of linear complementarity problem, which is useful to model several learning and optimization problems. In this letter, we propose an effective damped Newton method, PLS-DN, to find the exact (up to machine precision) solution of nondegenerate PLSs. PLS-DN exhibits provable semiiterative property, that is, the algorithm converges globally to the exact solution in a finite number of iterations. The rate of convergence is shown to be at least linear before termination. We emphasize the applications of our method in modeling, from a novel perspective of PLSs, some statistical learning problems such as box-constrained least squares, elitist Lasso (Kowalski & Torreesani, 2008), and support vector machines (Cortes & Vapnik, 1995). Numerical results on synthetic and benchmark data sets are presented to demonstrate the effectiveness and efficiency of PLS-DN on these problems.

Linear Complementarity Problem and Multiobjective Optimization

A new method is proposed for the linear complementarity problem (LCP). Firstly we formulate the LCP into a multiobjective optimization problem (MOP), and study the relations between the efficient solution of MOP and optimal solution of LCP. Based on the efficient solution of MOP, we define zero-efficient solution. Then we indicate that zero-efficient solution of the MOP is also the solution to the LCP. Finally some standard LCP examples are respectively transformed into MOP and solved by minimax method. Numerical results indicate that the proposed method is effective.

A FPTAS for computing a symmetric Leontief competitive economy equilibrium

In this paper, we consider a linear complementarity problem (LCP) arisen from the Nash and Arrow–Debreu competitive economy equilibria where the LCP coefficient matrix is symmetric. We prove that the decision problem, to decide whether or not there exists a complementary solution, is NP-complete. Under certain conditions, an LCP solution is guaranteed to exist and we present a fully polynomial-time approximation scheme (FPTAS) for approximating a complementary solution, although the LCP solution set can be non-convex or non-connected. Our method is based on approximating a quadratic social utility optimization problem (QP) and showing that a certain KKT point of the QP problem is an LCP solution. Then, we further show that such a KKT point can be approximated with a new improved running time complexity $${{O}((\frac{n^4}{\epsilon})\log\log(\frac{1}{\epsilon}))}$$ arithmetic operation in accuracy $${\epsilon \in (0,1)}$$. We also report preliminary computational results which show that the method is highly effective. Applications in competitive market model problems with other utility functions are also presented, including global trading and dynamic spectrum management problems.

Perturbation Bounds of P-Matrix Linear Complementarity Problems

We define a new fundamental constant associated with a P-matrix and show that this constant has various useful properties for the P-matrix linear complementarity problems (LCP). In particular, this constant is sharper than the Mathias-Pang constant in deriving perturbation bounds for the P-matrix LCP. Moreover, this new constant defines a measure of sensitivity of the solution of the P-matrix LCP. We examine how perturbations in the data affect the solution of the LCP and efficiency of Newton-type methods for solving the LCP.

Validation and enclosure of solutions of linear complementarity problems

An inclusion condition is presented to guarantee the existence of the solution of the linear complementarity problem in a given domain. The condition can be tested numerically with very small computational cost. Based on the condition algorithms are designed to compute an interval to enclose the unknown solution. Numerical results are reported to support the theoretical analysis in the paper.

On the Solution of Linear Complementarity Problem by A Stochastic Iteration Method

On the Solution of Linear Complementarity Problem by A Stochastic Iteration Method

On computer-assisted proofs for solutions of linear complementarity problems

In this paper we consider the linear complementarity problem where the components of the input data M and q are not exactly known but can be enclosed in intervals. We compare three tests to each other each of which can be used by a computer that supports interval arithmetic to give guaranteed bounds for a solution of the LCP defined by M and q.

A combined homotopy interior point method for the linear complementarity problem

A combined homotopy interior point method for the linear complementarity problem (LCP) is proposed in this paper. We construct a homotopy path for the linear complementarity problem and prove that the homotopy path determines a smooth interior path from an arbitrary given interior point to the solution of LCP. Furthermore, we modify a predictor corrector algorithm with the technique of the cone neighborhood to trace the homotopy path. Finally, a numerical example is provided to show the effectiveness and feasibility of this method.

Enclosing Solutions of Linear Complementarity Problems for H-Matrices

The paper establishes a computational enclosure of the solution of the linear complementarity problem (q, M), where M is assumed to be an H-matrix with a positive main diagonal. A class of problems with interval data, which can arise in approximating the solutions of free boundary problems, is also treated successfully.

Verification for existence of solutions of linear complementarity problems

Recently, G. Alefeld, X. Chen and F. Potra [Numer. Math. 83 (1999) 265–315] presented a verification method for solutions of linear complementarity problems (LCPs). This paper is an attempt to obtain more useful information from the output of this verification method. In particular, existing results can only claim the nonexistence of solutions in a given interval. We will use the Farkas lemma to check if the interval contains a negative certification which shows the nonexistence of solutions in the whole space R n . Moreover, we will study how to choose a good nonsingular matrix A in the interval operator for P 0 -matrix linear complementarity problems. We report numerical results to illustrate the efficiency of the proposed technique.

Superlinear Noninterior One-Step Continuation Method for Monotone LCP in the Absence of Strict Complementarity

We propose a noninterior continuation method for the monotone linear complementarity problem (LCP) by modifying the Burke–Xu framework of the noninterior predictor-corrector path-following method (Refs. 1–2). The new method solves one system of linear equations and carries out only one line search at each iteration. It is shown to converge to the LCP solution globally linearly and locally superlinearly without the assumption of strict complementarity at the solution. Our analysis of the continuation method is based on a broader class of the smooth functions introduced by Chen and Mangasarian (Ref. 3).

Um método de projeção para problemas de complementaridade linear

Este artigo apresenta um algoritmo de dois passos para a resolução numérica de problemas de complementaridade linear (LCP). O algoritmo que apresentaremos combina a estratégia de conjunto ativo aliada ao método dos gradientes conjugados quadrático para a resolução do sistema linear reduzido. O desenvolvimento deste método de dois passos foi baseada em um outro, também de dois passos, desenvolvido por Kocvara & Zowe (1994), que combina o método de super relaxação sucessiva simétrica com projeção e o método dos gradientes conjugados pré condicionado. Na seção Experiência Numérica, utilizaremos o método para resolver LCPs com matrizes não singulares pertencentes às classes de matrizes semi positivas definidas, P-matrizes e P0-matrizes. Será feita ainda uma comparação entre o desempenho numérico do algoritmo apresentado neste texto e o método desenvolvido por Pardalos, Ye, Han & Kaliski (1993).

Numerical validation of solutions of linear complementarity problems

This paper proposes a validation method for solutions of linear complementarity problems. The validation procedure consists of two sufficient conditions that can be tested on a digital computer. If the first condition is satisfied then a given multidimensional interval centered at an approximate solution of the problem is guaranteed to contain an exact solution. If the second condition is satisfied then the multidimensional interval is guaranteed to contain no exact solution. This study is based on the mean value theorem for absolutely continuous functions and the reformulation of linear complementarity problems as nonsmooth nonlinear systems of equations.

Stabilizing ill-conditioned linear complementarity problems

Least-squares adjustment with inequality constraints is equivalent to solving a linear complementarity problem (LCP) with a positive definite matrix; the latter has, however, received little attention in geodesy. Two kinds of LCP solution methods (direct and approximation) have been analysed from the point of view of solution stability. It has been found that they can result in an unreliable solution to an unstable LCP with a positive definite matrix, the extent of which depends on the corresponding submatrix. Several proposals for improving the solutions to the unstable LCP are suggested. Two examples are given.

Integral Solutions of Linear Complementarity Problems

We characterize the class of integral square matrices M having the property that for every integral vector q the linear complementarity problem with data M, q has only integral basic solutions. These matrices, called principally unimodular matrices, are those for which every principal nonsingular submatrix is unimodular. As a consequence, we show that if M is rank-symmetric and principally unimodular, and q is integral, then the problem has an integral solution if it has a solution. Principal unimodularity can be regarded as an extension of total unimodularity, and our results can be regarded as extensions of well-known results on integral solutions to linear programs. We summarize what is known about principally unimodular symmetric and skew-symmetric matrices.

On finite termination of an iterative method for linear complementarity problems

Based on a well-known reformulation of the linear complementarity problem (LCP) as a nondifferentiable system of nonlinear equations, a Newton-type method will be described for the solution of LCPs. Under certain assumptions, it will be shown that this method has a finite termination property, i.e., if an iterate is sufficiently close to a solution of LCP, the method finds this solution in one step. This result will be applied to a recently proposed algorithm by Harker and Pang in order to prove that their algorithm also has the finite termination property.