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.     

A path-following interior-point algorithm for linear and quadratic problems

We describe an algorithm for the monotone linear complementarity problem (LCP) that converges from any positive, not necessarily feasible, starting point and exhibits polynomial complexity if some additional assumptions are made on the starting point. If the problem has a strictly complementarity solution, the method converges subquadratically. We show that the algorithm and its convergence properties extend readily to the mixed monotone linear complementarity problem and, hence, to all the usual formulations of the linear programming and convex quadratic programming problems.This research was supported by the Office of Scientific Computing, U.S. Department of Energy, under Contract W-31-109-Eng-38.     

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.     

Fixed Point Methods for the Complementarity Problem

This paper is concerned with iterative procedures for the monotone complementarity problem. Our iterative methods consist of finding fixed points of appropriate continuous maps. In the case of the linear complementarity problem, it is shown that the problem is solvable if and only if the sequence of iterates is bounded in which case summability methods are used to find a solution of the problem. This procedure is then used to find a solution of the nonlinear complementarity problem satisfying certain regularity conditions for which the problem has a nonempty bounded solution set.     

An infeasible-interior-point predictor-corrector algorithm for theP *-geometric LCP

AP _*-geometric linear complementarity problem (P _*GP) as a generalization of the monotone geometric linear complementarity problem is introduced. In particular, it contains the monotone standard linear complementarity problem and the horizontal linear complementarity problem. Linear and quadratic programming problems can be expressed in a “natural” way (i.e., without any change of variables) asP _*GP. It is shown that the algorithm of Mizunoet al. [6] can be extended to solve theP _*GP. The extended algorithm is globally convergent and its computational complexity depends on the quality of the starting points. The algorithm is quadratically convergent for problems having a strictly complementary solution. The work of F. A. Potra was supported in part by NSF Grant DMS 9305760     

A full-Newton step infeasible interior-point algorithm for monotone LCP based on a locally-kernel function

We propose a new full-Newton step infeasible interior-point algorithm for monotone linear complementarity problems based on a simple locally-kernel function. The algorithm uses the simple locally-kernel function to determine the search directions and define the neighborhood of central path. Two types of full-Newton steps are used, feasibility step and centering step. The algorithm starts from strictly feasible iterates of a perturbed problem, on its central path, and feasibility steps find strictly feasible iterates for the next perturbed problem. By using centering steps for the new perturbed problem, we obtain strictly feasible iterates close enough to the central path of the new perturbed problem. The procedure is repeated until an ?-approximate solution is found. We analyze the algorithm and obtain the complexity bound, which coincides with the best-known result for monotone linear complementarity problems.     

Regularization Algorithms for Solving Monotone Ky Fan Inequalities with Application to a Nash-Cournot Equilibrium Model

We make use of the Banach contraction mapping principle to prove the linear convergence of a regularization algorithm for strongly monotone Ky Fan inequalities that satisfy a Lipschitz-type condition recently introduced by Mastroeni. We then modify the proposed algorithm to obtain a line search-free algorithm which does not require the Lipschitz-type condition. We apply the proposed algorithms to implement inexact proximal methods for solving monotone (not necessarily strongly monotone) Ky Fan inequalities. Applications to variational inequality and complementarity problems are discussed. As a consequence, a linearly convergent derivative-free algorithm without line search for strongly monotone nonlinear complementarity problem is obtained. Application to a Nash-Cournot equilibrium model is discussed and some preliminary computational results are reported.     

The Monotonicity of the Linear Complementarity Problem

The monotonicity of the linear complementarity problem (LCP) is discussed in this paper. Both the monotone property about the single element of the solution and the monotone property of the whole solution are presented. In order to illustrate the results, some corresponding numerical experiments are provided.     

A Lipschitzian error bound for monotone symmetric cone linear complementarity problem

We first discuss some properties of the solution set of a monotone symmetric cone linear complementarity problem (SCLCP), and then consider the limiting behaviour of a sequence of strictly feasible solutions within a wide neighbourhood of central trajectory for the monotone SCLCP. Under assumptions of strict complementarity and Slater’s condition, we provide four different characterizations of a Lipschitzian error bound for the monotone SCLCP in general Euclidean Jordan algebras. Thanks to the observation that a pair of primal-dual convex quadratic symmetric cone programming (CQSCP) problems can be exactly formulated as the monotone SCLCP, thus we obtain the same error bound results for CQSCP as a by-product.     

A Linearly Convergent Derivative-Free Descent Method for Strongly Monotone Complementarity Problems

We establish the first rate of convergence result for the class of derivative-free descent methods for solving complementarity problems. The algorithm considered here is based on the implicit Lagrangian reformulation [26, 35] of the nonlinear complementarity problem, and makes use of the descent direction proposed in [42], but employs a different Armijo-type linesearch rule. We show that in the strongly monotone case, the iterates generated by the method converge globally at a linear rate to the solution of the problem.     

ON THE MONOTONE CONVERGENCE OF THE PROJECTED ITERATION METHODS FOR LINEAR COMPLEMENTARITY PROBLEMS

Under suitable conditions,the monotone convergence about the projected iteration method for solving linear complementarity problem is proved and the influence of the involved parameter matrix on the convergence rate of this method is investigated.     

Global linear convergence of a path-following algorithm for some monotone variational inequality problems

We consider a primal-scaling path-following algorithm for solving a certain class of monotone variational inequality problems. Included in this class are the convex separable programs considered by Monteiro and Adler and the monotone linear complementarity problem. This algorithm can start from any interior solution and attain a global linear rate of convergence with a convergence ratio of 1 ?c/√m, wherem denotes the dimension of the problem andc is a certain constant. One can also introduce a line search strategy to accelerate the convergence of this algorithm.     

一个求解水平线性互补问题的高阶可行内点算法的多项式复杂性

最近,Zhao和Sun提出了一个求解sufficient线性互补问题的高阶不可行内点算法.不需要严格互补解条件,他们的算法获得了高阶局部收敛率,但他们的文章没有报告多项式复杂性结果.本文我们考虑他们所给算法的一个简化版本,即考虑求解单调水平线性互补问题的一个高阶可行内点算法.我们证明了算法的迭代复杂性是     

非线性互补问题的一种不可行非内点连续算法

基于Chen-Harker—Kanzow-Smale光滑函数，对单调非线性互补问题NCP(f)给出了一种不可行非内点连续算法，该算法在每次迭代时只需求解一个线性等式系统，执行一次线搜索，算法在NCP(f)的解处不需要严格互补的条件下，具有全局线性收敛性和局部二次收敛性．     

Error bounds and strong upper semicontinuity for monotone affine variational inequalities

Global error bounds for possibly degenerate or nondegenerate monotone affine variational inequality problems are given. The error bounds are on an arbitrary point and are in terms of the distance between the given point and a solution to a convex quadratic program. For the monotone linear complementarity problem the convex program is that of minimizing a quadratic function on the nonnegative orthant. These bounds may form the basis of an iterative quadratic programming procedure for solving affine variational inequality problems. A strong upper semicontinuity result is also obtained which may be useful for finitely terminating any convergent algorithm by periodically solving a linear program.This material is based on research supported by Air Force Office of Scientific Research Grant AFOSR-89-0410 and National Science Foundation Grants CCR-9101801 and CCR-9157632.     

Stable monotone variational inequalities

Variational inequalities associated with monotone operators (possibly nonlinear and multivalued) and convex sets (possibly unbounded) are studied in reflexive Banach spaces. A variety of results are given which relate to a stability concept involving a natural parameter. These include characterizations useful as criteria for stable existence of solutions and also several characterizations of surjectivity. The monotone complementarity problem is covered as a special case, and the results are sharpened for linear monotone complementarity and for generalized linear programming.Sponsored by the United States Army under Contract No. DAAG29-80-C-0041 at the University of Wisconsin - Madison and by the National Science Foundation under Grant No. DMS-8405179 at the University of Illinois at Urbana-Champaign.     

求解退化单调线性互补问题极大互补解的复杂性

本文考虑求解退化单调线性互补问题的一类不可行内点算法，其中嵌入一个恢复算法，给出了用这类算法产生所考虑问题的一个精确极大互补解的复杂性．     

A Polynomial Interior-Point Algorithm for Monotone Linear Complementarity Problems

In this paper, we propose an interior-point algorithm for monotone linear complementarity problems. The algorithm is based on a new technique for finding the search direction and the strategy of the central path. At each iteration, we use only full-Newton steps. Moreover, it is proven that the number of iterations of the algorithm coincides with the well-known best iteration bound for monotone linear complementarity problems.     

Complementarity problems over cones with monotone and pseudomonotone maps

The notion of a monotone map is generalized to that of a pseudomonotone map. It is shown that a differentiable, pseudoconvex function is characterized by the pseudomonotonicity of its gradient. Several existence theorems are established for a given complementarity problem over a certain cone where the underlying map is either monotone or pseudomonotone under the assumption that the complementarity problem has a feasible or strictly feasible point.This work was supported in part by the National Science Foundation, Grant No. GP-34619.     

On the regularization of mixed complementarity problems

A variational inequality problem (VIP) satisfying a constraint qualification can be reduced to a mixed complementarity problem (MOP). Monotonicity of the VIP implies that the MOP is also monotone. Introducing regularizing perturbations, a sequence of strictly monotone mixed complementarity problems is generated. It is shown that, if the original problem is solvable, the sequence of computable inexact solutions of the strictly monotone MCP's is bounded and every accumulation point is a solution. Under an additional condition on the precision used for solving each subproblem, the sequence converges to the minimum norm solution of the MCP.