Conditions for Error Bounds of Linear Complementarity Problems over Second-Order Cones with Pseudomonotonicity

We first discuss some properties of the solution set of a pseudomonotone second-order cone linear complementarity problem (SOCLCP), and then analyse the limiting behavior of a sequence of strictly feasible solutions within a new wide neighborhood of the central trajectory for the pseudomonotone SOCLCP under assumptions of strict complementarity. Based on this, we derive four different characterizations of an error bound for the pseudomonotone SOCLCP.     

A proximal gradient descent method for the extended second-order cone linear complementarity problem

We consider an extended second-order cone linear complementarity problem (SOCLCP), including the generalized SOCLCP, the horizontal SOCLCP, the vertical SOCLCP, and the mixed SOCLCP as special cases. In this paper, we present some simple second-order cone constrained and unconstrained reformulation problems, and under mild conditions prove the equivalence between the stationary points of these optimization problems and the solutions of the extended SOCLCP. Particularly, we develop a proximal gradient descent method for solving the second-order cone constrained problems. This method is very simple and at each iteration makes only one Euclidean projection onto second-order cones. We establish global convergence and, under a local Lipschitzian error bound assumption, linear rate of convergence. Numerical comparisons are made with the limited-memory BFGS method for the unconstrained reformulations, which verify the effectiveness of the proposed method.     

The GUS-property of second-order cone linear complementarity problems

The globally uniquely solvable (GUS) property of the linear transformation of the linear complementarity problems over symmetric cones has been studied recently by Gowda et al. via the approach of Euclidean Jordan algebra. In this paper, we contribute a new approach to characterizing the GUS property of the linear transformation of the second-order cone linear complementarity problems (SOCLCP) via some basic linear algebra properties of the involved matrix of SOCLCP. Some more concrete and checkable sufficient and necessary conditions for the GUS property are thus derived.     

A generalized Newton method for absolute value equations associated with second order cones

In this paper, we introduce the absolute value equations associated with second order cones (SOCAVE in short), which is a generalization of the absolute value equations discussed recently in the literature. It is proved that the SOCAVE is equivalent to a class of second order cone linear complementarity problems (SOCLCP in short). In particular, we propose a generalized Newton method for solving the SOCAVE and show that the proposed method is globally linearly and locally quadratically convergent under suitable assumptions. We also report some preliminary numerical results of the proposed method for solving the SOCAVE and the SOCLCP, which show the efficiency of the proposed method.     

Solution analysis for the pseudomonotone second-order cone linear complementarity problem

In this paper, we are concerned with the set of the solutions and the geometric property of the pseudomonotone second-order cone linear complementarity problems (SOCLCP). Based on Tao’s recent work [Tao, J. Optim. Theory Appl., 159, 41–56 (2013)] on pseudomonotone LCP on Euclidean Jordan algebras, we characterize the set of solutions and also derive intrinsic properties that reveal the underlying geometry of the pseudomonotone SOCLCP.     

The basic theorem of complementarity revisited

The basic theorm of (linear) complementarity was stated in a 1971 paper [6] by B.C. Eaves who credited C.E. Lemke for giving a constructive proof based on his almost complementary pivot algorithm. This theorem asserts that associated with an arbitrary linear complementarity problem, a certain augmented problem always possesses a solution. Many well-known existence results pertaining to the linear complementarity problem are consequences of this fundamental theorem.In this paper, we explore some further implications of the basic theorem of complementarity and derive new existence results for the linear complementarity problem. Based on these results, conditions for the existence of a solution to a linear complementarity problem with a fully-semimonotone matrix are examined. The class of the linear complementarity problems with aG-matrix is also investigated.The work of this author was based on research supported by the National Science Foundation under grant ECS-8717968.     

Equivalence of the Generalized Vertical Block Linear Complementarity Problems and Linear Complementarity Problems

In this paper, generalization of a vertical block linear complementarity problem associated with two different types of matrices, one of which is a square matrix and the other is a vertical block matrix, is proposed. The necessary and sufficient conditions for the existence of the solution of the generalized vertical block linear complementarity problem is derived and the relationship between the solution set of the generalized vertical block linear complementarity problem and the linear complementarity problem is established. It is proved that the generalized vertical block linear complementarity problem has the P-property if and only if the vertical block linear complementarity problem has the P-property.     

Nonmonotone Inexact Newton Method for the Extended Linear Complementarity Problem

This paper presents a nonmonotone inexact Newton-type method for the extended linear complementarity problem (ELCP). We first reformulate the optimization system of the ELCP problem into a system of smoothed equations. Then we solve this system by a nonmonotone inexact Newton-type algorithm. The global convergence is obtained and numerical tests for some classes of ELCP include linear complementarity, horizontal linear complementarity, and generalized linear complementarity problems are also given to show the e?ciency of the proposed algorithm.     

A method for solving the general parametric linear complementarity problem

This paper presents a solution method for the general (mixed integer) parametric linear complementarity problem pLCP(q(θ),M), where the matrix M has a general structure and integrality restriction can be enforced on the solution. Based on the equivalence between the linear complementarity problem and mixed integer feasibility problem, we propose a mixed integer programming formulation with an objective of finding the minimum 1-norm solution for the original linear complementarity problem. The parametric linear complementarity problem is then formulated as multiparametric mixed integer programming problem, which is solved using a multiparametric programming algorithm. The proposed method is illustrated through a number of examples.     

The generalized linear complementarity problem revisited

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.     

改进的分块模方法求解对角占优线性互补问题

为了高效求解中小型线性互补问题,本文提出了改进的分块模方法,并证明了关于严格对角占优(对角元素均为正数)线性互补问题的收敛性.对于广义对角占优线性互补问题,先将其转化为严格对角占优线性互补问题,再采用改进的分块模方法求解.数值结果表明,改进的分块模方法在求解广义对角占优线性互补问题时在内迭代次数和计算时间上均明显优于分...     

垂直线性互补问题的一步全局线性和局部二次收敛光滑Newton法

基于凝聚函数,提出一个求解垂直线性互补问题的光滑Newton法.该算法具有以下优点:(i)每次迭代仅需解一个线性系统和实施一次线性搜索;(ⅱ)算法对垂直分块P0矩阵的线性互补问题有定义且迭代序列的每个聚点都是它的解.而且,对垂直分块P₀+R₀矩阵的线性互补问题,算法产生的迭代序列有界且其任一聚点都是它的解;(ⅲ)在无严格互补条件下证得算法即具有全局线性收敛性又具有局部二次收敛性.许多已存在的求解此问题的光滑Newton法都不具有性质(ⅲ).     

Two characterization theorems in complementarity theory

In this paper, we establish two characterization theorems for the linear and non-linear complementarity problems. Theorem 1 concerns the local uniqueness of a solution to a linear complementarity problem. Theorem 2 provides a necessary and sufficient condition for the differentiability of a solution to a parametric non-linear complementarity problem.     

解线性互补问题的预处理加速模Gauss-Seidel迭代方法

本文提出了解线性互补问题的预处理加速模系Gauss-Seidel迭代方法,当线性互补问题的系统矩阵是M-矩阵时证明了方法的收敛性,并给出了该预处理方法关于原方法的一个比较定理.数值实验显示该预处理迭代方法明显加速了原方法的收敛.     

Nonlinear mappings associated with the generalized linear complementarity problem

We show that the Cottle—Dantzig generalized linear complementarity problem (GLCP) is equivalent to a nonlinear complementarity problem (NLCP), a piecewise linear system of equations (PLS), a multiple objective programming problem (MOP), and a variational inequalities problem (VIP). On the basis of these equivalences, we provide an algorithm for solving problem GLCP.Project partially supported by a grant from Oak Ridge Associated Universities, TN, USA.     

A Smoothing Method for a Mathematical Program with P-Matrix Linear Complementarity Constraints

We consider a mathematical program whose constraints involve a parametric P-matrix linear complementarity problem with the design (upper level) variables as parameters. Solutions of this complementarity problem define a piecewise linear function of the parameters. We study a smoothing function of this function for solving the mathematical program. We investigate the limiting behaviour of optimal solutions, KKT points and B-stationary points of the smoothing problem. We show that a class of mathematical programs with P-matrix linear complementarity constraints can be reformulated as a piecewise convex program and solved through a sequence of continuously differentiable convex programs. Preliminary numerical results indicate that the method and convex reformulation are promising.     

Feasible smooth method based on Barzilai–Borwein method for stochastic linear complementarity problem

In this paper, we propose a feasible smooth method based on Barzilai–Borwein (BB) for stochastic linear complementarity problem. It is based on the expected residual minimization (ERM) formulation for the stochastic linear complementarity problem. Numerical experiments show that the method is efficient.     

On Linear Fractional Programming Problem and its Computation Using a Neural Network Model

In this paper we consider linear fractional programming problem and look at its linear complementarity formulation. In the literature, uniqueness of solution of a linear fractional programming problem is characterized through strong quasiconvexity. We present another characterization of uniqueness through complementarity approach and show that the solution set of a fractional programming problem is convex. Finally we formulate the complementarity condition as a set of dynamical equations and prove certain results involving the neural network model. A computational experience is also reported.      

Solution of Complementarity Problems via Piecewise Linearization

My master thesis concerns the solution linear complementarity problems (LCP). The Lemke algorithm, the most commonly used algorithm for solving a LCP until this day, was compared with the piecewise Newton method (PLN algorithm). The piecewise Newton method is an algorithm to solve a piecewise linear system on the basis of damped Newton methods. The linear complementarity problem is formulated as a piecewise linear system for the applicability of the PLN algorithm. Then, different application examples will be presented, solved with the PLN algorithm. As a result of the findings (of my master thesis) it can be assumed that – under the condition of coherent orientation – the PLN-algorithm requires fewer iterations to solve a linear complementarity problem than the Lemke algorithm. The coherent orientation for piecewise linear problems corresponds for linear complementarity problems to the P-matrix-property. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the solution and complexity of a generalized linear complementarity problem

We introduce some sufficient conditions under which a generalized linear complementarity problem (GLCP) can be solved as a pure linear complementarity problem. We also establish that the GLCP is in general a NP-Hard problem.Support of this work has been provided by the Instituto Nacional de Investigação Cientifica de Portugal (INIC) under contract 89/EXA/5.