Nonconvergence of the plain Newton-min algorithm for linear complementarity problems with a P-matrix

The plain Newton-min algorithm to solve the linear complementarity problem (LCP for short) can be viewed as a semismooth Newton algorithm without globalization technique to solve the system of piecewise linear equations min(x, Mx + q) = 0, which is equivalent to the LCP. When M is an M-matrix of order n, the algorithm is known to converge in at most n iterations. We show in this paper that this result no longer holds when M is a P-matrix of order ≥ 3, since then the algorithm may cycle. P-matrices are interesting since they are those ensuring the existence and uniqueness of the solution to the LCP for an arbitrary q. Incidentally, convergence occurs for a P-matrix of order 1 or 2.     

Randomized pivot algorithms for P-matrix linear complementarity problems

We study orientations of the n-cube that come from simple principal pivot algorithms for the linear complementarity problem with a P-matrix. We show that these orientations properly generalize those that are obtained from linear objective functions on polytopes combinatorially equivalent to the cube. The orientations from LCP with a P-matrix may admit directed cycles. We give a sequence of problems on which the algorithm RANDOM-EDGE performs very badly. Received: February 12, 2001 / Accepted: September 9, 2001?Published online April 12, 2002     

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.     

An investigation of interior-point and block pivoting algorithms for large-scale symmetric monotone linear complementarity problems

In this paper we describe a computational study of block principal pivoting (BP) and interior-point predictor-corrector (PC) algorithms for the solution of large-scale linear complementarity problems (LCP) with symmetric positive definite matrices. This study shows that these algorithms are in general quite appropriate for this type of LCPs. The BP algorithm does not seem to be sensitive to bad scaling and degeneracy of the unique solution of the LCP, while these aspects have some effect on the performance of the PC algorithm. On the other hand, the BP method has not performed well in two LCPs with ill-conditioned matrices for which the PC algorithm has behaved quite well.A hybrid algorithm combining these two techniques is also introduced and seems to be the most robust procedure for the solution of large-scale LCPs with symmetric positive definite matrices.Support of this work has been provided by the Instituto de Telecomunicações.     

Characterizing Q-Matrices beyond L-Matrices

The characterization of Q-matrices, within the class of P₀ matrices due to Aganagič and Cottle, is well known. Afterward, Pang proved a similar characterization for the class L which does not contain class P₀. In this note, we establish furher the same result of Pang for a new class of matrices which properly contains class L. Furthermore, the equivalence between a Q-matrix and a Q_b-matrix, which consists of matrices such that the linear complementarity problem LCP(q,M) has a nonempty and compact solution set for all , is discussed within such a new class. Positive subdefinite matrices with rank one are specially analyzed. This work was supported by CONICYT-Chile through FONDECYT 104-0610, FONDAP-Matemáticas Aplicadas II, and Proyecto MECESUP UCO 9907.     

Two class of synchronous matrix multisplitting schemes for solving linear complementarity problems

Many problems in the areas of scientific computing and engineering applications can lead to the solution of the linear complementarity problem LCP (M,q). It is well known that the matrix multisplitting methods have been found very useful for solving LCP (M,q). In this article, by applying the generalized accelerated overrelaxation (GAOR) and the symmetric successive overrelaxation (SSOR) techniques, we introduce two class of synchronous matrix multisplitting methods to solve LCP (M,q). Convergence results for these two methods are presented when M is an H-matrix (and also an M-matrix). Also the monotone convergence of the new methods is established. Finally, the numerical results show that the introduced methods are effective for solving the large and sparse linear complementary problems.     

OnQ-matrices

In a recent paper [1], Aganagic and Cottle have established a constructive characterization for aP ₀-matrix to be aQ-matrix. Among the principal results in this paper, we show that the same characterization holds for anL-matrix as well, and that the symmetric copositive-plusQ-matrices are precisely those which are strictly copositive.     

New multiplier method for solving linear complementarity problems

A new multiplier method for solving the linear complementarity problem LCP(q, M) is proposed. By introducing a Lagrangian of LCP(q, M), a new smooth merit function ϑ(x, λ) for LCP(q, M) is constructed. Based on it, a simple damped Newton-type algorithm with multiplier self-adjusting step is presented. When M is a P-matrix, the sequence {ϑ(x ^k, λ ^k)} (where {(x ^k, λ ^k)} is generated by the algorithm) is globally linearly convergent to zero and convergent in a finite number of iterations if the solution is degenerate. Numerical results suggest that the method is highly efficient and promising. Selected from Numerical Mathematics (A Journal of Chinese Universities), 2004, 26(2): 162–171     

Sub-quadratic convergence of a smoothing Newton algorithm for the <Emphasis Type="Italic">P</Emphasis><Subscript><Emphasis Type="Italic">0</Emphasis></Subscript>– and monotone LCP

Given , the linear complementarity problem (LCP) is to find such that (x, s) 0,s=Mx+q,x^Ts=0. By using the Chen-Harker-Kanzow-Smale (CHKS) smoothing function, the LCP is reformulated as a system of parameterized smooth-nonsmooth equations. As a result, a smoothing Newton algorithm, which is a modified version of the Qi-Sun-Zhou algorithm [Mathematical Programming, Vol. 87, 2000, pp. 1–35], is proposed to solve the LCP with M being assumed to be a P₀-matrix (P₀–LCP). The proposed algorithm needs only to solve one system of linear equations and to do one line search at each iteration. It is proved in this paper that the proposed algorithm has the following convergence properties: (i) it is well-defined and any accumulation point of the iteration sequence is a solution of the P₀–LCP; (ii) it generates a bounded sequence if the P₀–LCP has a nonempty and bounded solution set; (iii) if an accumulation point of the iteration sequence satisfies a nonsingularity condition, which implies the P₀–LCP has a unique solution, then the whole iteration sequence converges to this accumulation point sub-quadratically with a Q-rate 2–t, where t(0,1) is a parameter; and (iv) if M is positive semidefinite and an accumulation point of the iteration sequence satisfies a strict complementarity condition, then the whole sequence converges to the accumulation point quadratically.This authors work is supported by the Hong Kong Research Grant Council and the Australian Research Council.This authors work is supported by Grant R146-000-035-101 of National University of Singapore.Mathematics Subject Classification (1991): 90C33, 65K10     

Polar reciprocal convex bodies

The minimum of the product of the volume of a symmetric convex bodyK and the volume of the polar reciprocal body ofK relative to the center of symmetry is attained for the cube and then-dimensional crossbody. As a consequence, there is a sharp upper bound in Mahler’s theorem on successive minima in the geometry of numbers. The difficulties involved in the determination of the minimum for unsymmetricK are discussed. Reserch partially supported by NSF Grant GP-27960. An erratum to this article is available at .     

IGAOR and multisplitting IGAOR methods for linear complementarity problems

In this paper, we propose an interval version of the generalized accelerated overrelaxation methods, which we refer to as IGAOR, for solving the linear complementarity problems, LCP (M, q), and develop a class of multisplitting IGAOR methods which can be easily implemented in parallel. In addition, in regards to the H-matrix with positive diagonal elements, we prove the convergence of these algorithms and illustrate their efficiency through our numerical results.     

Semidifferentiable functions and necessary optimality conditions

We consider Lyapunov's equationPA+A ^T P+Q=0, whereQ is symmetric positive definite andA is in controllable companion form. We prove that a necessary and sufficient condition thatA be stable is that the first rowP ₁ of theP-matrix be a stablen–1 coefficient vector. This result is related to the minimum phase property of linear systems and is useful in designing robust controllers.     

On the convergence of the symmetric SOR method for matrices with red-black ordering

Summary We prove that if the matrixA has the structure which results from the so-called red-black ordering and ifA is anH-matrix then the symmetric SOR method (called the SSOR method) is convergent for 0<<2. In the special case thatA is even anM-matrix we show that the symmetric single-step method cannot be accelerated by the SSOR method. Symmetry of the matrixA is not assumed.     

Computational complexity of LCPs associated with positive definite symmetric matrices

Murty in a recent paper has shown that the computational effort required to solve a linear complementarity problem (LCP), by either of the two well known complementary pivot methods is not bounded above by a polynomial in the size of the problem. In that paper, by constructing a class of LCPs—one of ordern forn 2—he has shown that to solve the problem of ordern, either of the two methods goes through 2 ⁿ pivot steps before termination.However that paper leaves it as an open question to show whether or not the same property holds if the matrix,M, in the LCP is positive definite and symmetric. The class of LCPs in whichM is positive definite and symmetric is of particular interest because of the special structure of the problems, and also because they appear in many practical applications.In this paper, we study the computational growth of each of the two methods to solve the LCP, (q, M), whenM is positive definite and symmetric and obtain similar results.This research is partially supported by Air Force Office of Scientific Research, Air Force Number AFOSR-78-3646. The United States Government is authorized to reproduce and distribute reprints for governmental purposes notwithstanding any copyright notation thereon.     

Growth behavior of a class of merit functions for the nonlinear complementarity problem

When the nonlinear complementarity problem is reformulated as that of finding the zero of a self-mapping, the norm of the selfmapping serves naturally as a merit function for the problem. We study the growth behavior of such a merit function. In particular, we show that, for the linear complementarity problem, whether the merit function is coercive is intimately related to whether the underlying matrix is aP-matrix or a nondegenerate matrix or anR _o-matrix. We also show that, for the more popular choices of the merit function, the merit function is bounded below by the norm of the natural residual raised to a positive integral power. Thus, if the norm of the natural residual has positive order of growth, then so does the merit function.This work was partially supported by the National Science Foundation Grant No. CCR-93-11621.The author thanks Dr. Christian Kanzow for his many helpful comments on a preliminary version of this paper. He also thanks the referees for their helpful suggestions.     

Factorization of symmetric singular M-matrices

Suppose A is a symmetric, singular M-matrix. A sufficient condition for A to have a triangular, singular M-matrix factorization is given, and it is shown that PAP^T always has such a factorization for a particular permutation matrix P.     

High Order Long-Step Methods for Solving Linear Complementarity Problems

The paper is concerned with methods for solving linear complementarity problems (LCP) that are monotone or at least sufficient in the sense of Cottle, Pang and Venkateswaran (1989). A basic concept of interior-point-methods is the concept of (perhaps weighted) feasible or infeasible interior-point paths. They converge to a solution of the LCP if a natural path parameter, usually the current duality gap, tends to 0.After reviewing some basic analyticity properties of these paths it is shown how these properties can be used to devise also long-step path-following methods (and not only predictor–corrector type methods) for which the duality gap converges Q-superlinearly to 0 with an arbitrarily high order.     

The Compression Property for Affine Variational Inequalities

In 1965, Gale and Nikaidô showed that for any n × n P-matrix A, the only nonnegative vector that A sends into a nonpositive vector is the origin. They applied that result to derive various results including univalence properties of certain nonlinear functions. In this article, we show that an extension of their result holds with the nonnegative orthant replaced by any nonempty polyhedral convex cone. In place of the P-matrix condition, we require a determinantal condition that we call the compression property. When the polyhedral convex cone is the nonnegative orthant, the compression property reduces to the property of being a P-matrix and we recover the Gale-Nikaidô result. We apply the extended theorem to derive tools useful in the analysis of affine variational inequalities over polyhedral convex cones.     

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.This material is based upon work supported by the National Science Foundation under Grant No. DMII-9121419 to the first author and Grant No. DMII-9114489 to the third author. The authors gratefully acknowledge the constructive suggestions of a referee that helped focus the approach and its presentation.     

On Eisenbud's and Wigner's R-matrix: A general approach

The main objective of this paper is to give a rigorous treatment of Wigner's and Eisenbud's R-matrix method for scattering matrices of scattering systems consisting of two selfadjoint extensions of the same symmetric operator with finite deficiency indices. In the framework of boundary triplets and associated Weyl functions an abstract generalization of the R-matrix method is developed and the results are applied to Schrödinger operators on the real axis.