The solution set structure of monotone linear complementarity problems over second-order cone

The second-order cone linear complementarity problem (SOCLCP) is a generalization of the linear complementarity problem (LCP). In this paper we characterize the solution set of a monotone SOCLCP with the help of the Jordan-algebraic technique.     

On the solution of NP-hard linear complementarity problems

In this paper two enumerative algorithms for the Linear Complementarity Problems (LCP) are discussed. These procedures exploit the equivalence of theLCP into a nonconvex quadratic and a bilinear programs. It is shown that these algorithms are efficient for processing NP-hardLCPs associated with reformulations of the Knapsack problem and should be recommended to solve difficultLCPs.     

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.     

Robust solution of monotone stochastic linear complementarity problems

We consider the stochastic linear complementarity problem (SLCP) involving a random matrix whose expectation matrix is positive semi-definite. We show that the expected residual minimization (ERM) formulation of this problem has a nonempty and bounded solution set if the expected value (EV) formulation, which reduces to the LCP with the positive semi-definite expectation matrix, has a nonempty and bounded solution set. We give a new error bound for the monotone LCP and use it to show that solutions of the ERM formulation are robust in the sense that they may have a minimum sensitivity with respect to random parameter variations in SLCP. Numerical examples including a stochastic traffic equilibrium problem are given to illustrate the characteristics of the solutions.     

An algorithm for the fast solution of symmetric linear complementarity problems

This paper studies algorithms for the solution of mixed symmetric linear complementarity problems. The goal is to compute fast and approximate solutions of medium to large sized problems, such as those arising in computer game simulations and American options pricing. The paper proposes an improvement of a method described by Kocvara and Zowe (Numer Math 68:95–106, 1994) that combines projected Gauss–Seidel iterations with subspace minimization steps. The proposed algorithm employs a recursive subspace minimization designed to handle severely ill-conditioned problems. Numerical tests indicate that the approach is more efficient than interior-point and gradient projection methods on some physical simulation problems that arise in computer game scenarios. The research of J. L. Morales was supported by Asociación Mexicana de Cultura AC and CONACyT-NSF grant J110.388/2006. The research of J. Nocedal was supported by National Science Foundation grant CCF-0514772, Department of Energy grant DE-FG02-87ER25047-A004 and a grant from the Intel Corporation.     

On the connectedness of the solution set to linear complementarity systems

This paper establishes sufficient conditions for the connectedness of nontrivial subsets of the solution set to linear complementarity systems with special structure. Connectedness may be important to investigate stability and sensitivity questions, parametric problems, and for extending a Lemke-type method to a new class of problems. Such a property may help in analyzing the structure of the feasible region by checking the explicitly given matrices of the resulting conditions. From the point of view of geometry, the question is how to analyze the combined geometrical object consisting of a Riemannian manifold, a pointed cone, and level sets determined by linear inequalities.This paper has been mainly prepared while the author was visiting the Department of Mathematics at the University of Pisa. This research was partialy supported by the Hungarian National Research Foundation, Grant No. OTKA-2568.     

Generalized linear complementarity problems

It has been shown by Lemke that if a matrix is copositive plus on ⁿ , then feasibility of the corresponding linear complementarity problem implies solvability. In this article we show, under suitable conditions, that feasibility of ageneralized linear complementarity problem (i.e., defined over a more general closed convex cone in a real Hilbert space) implies solvability whenever the operator is copositive plus on that cone. We show that among all closed convex cones in a finite dimensional real Hilbert Space, polyhedral cones are theonly ones with the property that every copositive plus, feasible GLCP is solvable. We also prove a perturbation result for generalized linear complementarity problems.This research has been partially supported by the Air Force Office of Scientific Research under grants #AFOSR-82-0271 and #AFOSR-87-0350.     

Newton's method for linear complementarity problems

This paper presents an iterative, Newton-type method for solving a class of linear complementarity problems. This class was discovered by Mangasarian who had established that these problems can be solved as linear programs. Cottle and Pang characterized solutions of the problems in terms of least elements of certain polyhedral sets. The algorithms developed in this paper are shown to converge to the least element solutions. Some applications and computational results are also discussed.     

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.This work was partially supported by the National Science Foundation under grant number MCS 77-27632.     

The optimal solution set of the interval linear programming problems

Several methods exist for solving the interval linear programming (ILP) problem. In most of these methods, we can only obtain the optimal value of the objective function of the ILP problem. In this paper we determine the optimal solution set of the ILP as the intersection of some regions, by the best and the worst case (BWC) methods, when the feasible solution components of the best problem are positive. First, we convert the ILP problem to the convex combination problem by coefficients 0 ≤ λ _j , μ _ij , μ _i  ≤ 1, for i = 1, 2, . . . , m and j = 1, 2, . . . , n. If for each i, j, μ _ij  = μ _i  = λ _j  = 0, then the best problem has been obtained (in case of minimization problem). We move from the best problem towards the worst problem by tiny variations of λ _j , μ _ij and μ _i from 0 to 1. Then we solve each of the obtained problems. All of the optimal solutions form a region that we call the optimal solution set of the ILP. Our aim is to determine this optimal solution set by the best and the worst problem constraints. We show that some theorems to validity of this optimal solution set.     

Chaotic iterative methods for the linear complementarity problems

A class of parallel multisplitting chaotic relaxation methods is established for the large sparse linear complementarity problems, and the global and monotone convergence is proved for the H-matrix and the L-matrix classes, respectively. Moreover, comparison theorem is given, which describes the influences of the parameters and the multiple splittings upon the monotone convergence rates of the new methods.     

A sign-based linear method for horizontal linear complementarity problems

Numerical Algorithms - For the horizontal linear complementarity problem, we establish a linear method based on the sign patterns of the solution of the equivalent modulus equation under the...     

Robustness and nondegenerateness for linear complementarity problems

In this paper the main focus is on a stability concept for solutions of a linear complementarity problem. A solution of such a problem is robust if it is stable against slight perturbations of the data of the problem. Relations are investigated between the robustness, the nondegenerateness and the isolatedness of solutions. It turns out that an isolated nondegenerate solution is robust and also that a robust nondegenerate solution is isolated. Since the class of linear complementarity problems with only robust solutions or only nondegenerate solutions is not an open set, attention is paid to Garcia's classG _n of linear complementarity problems. The nondegenerate problems inG _n form an open set.     

An infeasible-interior-point algorithm for linear complementarity problems

We modify the algorithm of Zhang to obtain anO(n²L) infeasible-interior-point algorithm for monotone linear complementarity problems that has an asymptoticQ-subquadratic convergence rate. The algorithm requires the solution of at most two linear systems with the same coefficient matrix at each iteration.This research was supported by the Office of Scientific Computing, U.S. Department of Energy, under Contract W-31-109-Eng-38.     

Interior point methods for linear complementarity problems

A short survey of interior point methods for linear complementarity problems with emphasis on algorithms that have both polynomial complexity and superlinear convergence is presented. Some recent results obtained by the author and his collaborators are briefly summarized and several directions of future research are proposed. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Error bounds for monotone linear complementarity problems

We give a bound on the distance between an arbitrary point and the solution set of a monotone linear complementarity problem in terms of a condition constant that depends on the problem data only and a residual function of the violations of the complementary problem conditions by the point considered. When the point satisfies the linear inequalities of the complementarity problem, the residual consists of the complementarity condition plus its square root. This latter term is essential and without it the error bound cannot hold. We also show that another natural residual that has been employed to bound errors for strictly monotone linear complementarity problems fails to bound errors for the monotone case considered here. Sponsored by the United States Army under contract No. DAAG29-80-C-0041. This material is based on research sponsored by National Foundation Grant DCR-8420963 and Air Force Office of Scientific Research Grant AFOSR-ISSA-85-00080.     

Inverse eigenvalue problems for linear complementarity systems

Traditionally an inverse eigenvalue problem is about reconstructing a matrix from a given spectral data. In this work we study the set of real matrices A of order n such that the linear complementarity system

     

Corrector-predictor methods for sufficient linear complementarity problems

We present a new corrector-predictor method for solving sufficient linear complementarity problems for which a sufficiently centered feasible starting point is available. In contrast with its predictor-corrector counterpart proposed by Miao, the method does not depend on the handicap κ of the problem. The method has \(O((1+\kappa)\sqrt{n}L)\)-iteration complexity, the same as Miao’s method, but our error estimates are sightly better. The algorithm is quadratically convergent for problems having a strictly complementary solution. We also present a family of infeasible higher order corrector-predictor methods that are superlinearly convergent even in the absence of strict complementarity. The algorithms of this class are globally convergent for general positive starting points. They have \(O((1+\kappa)\sqrt{n}L)\)-iteration complexity for feasible, or “almost feasible”, starting points and O((1+κ)² nL)-iteration complexity for “sufficiently large” infeasible starting points.     

Inexact multisplitting methods for linear complementarity problems

We present an inexact multisplitting method for solving the linear complementarity problems, which is based on the inexact splitting method and the multisplitting method. This new method provides a specific realization for the multisplitting method and generalizes many existing matrix splitting methods for linear complementarity problems. Convergence for this new method is proved when the coefficient matrix is an _H+$H_{+}$-matrix. Then, two specific iteration forms for this inexact multisplitting method are presented, where the inner iterations are implemented either through a matrix splitting method or through a damped Newton method. Convergence properties for both these specific forms are analyzed, where the system matrix is either an _H+$H_{+}$-matrix or a symmetric matrix.