. The nonmonotone linear complementarity problem (LCP) is formulated as a bilinear program with separable constraints and an objective function that minimizesa natural error residual for the LCP. A linear-programming-basedalgorithm applied to the bilinear program terminates in a finite number of steps at a solution or stationary point of the problem. The bilinear algorithm solved 80 consecutive cases of the LCP formulation of the knapsack feasibility problem ranging in size between 10 and 3000, with almost constant average number of major iterations equal to four. Keywords: linear complementarity, bilinear programming, knapsack 1. Introduction It is well known that the linear complementarity problem [4], [16] 0 x ? Mx+ q 0; (1) for a given n \Theta n real matrix M and a given n \Theta 1 vector q, can be written as the bilinear program min x;w fx 0 wjw = Mx+ q; x 0; w 0g: (2) For the case of a general M , considered here, the objective function of (2) is nonconvex and the cons...

This paper proposes a validation method for solutions of linear complementarity problems. The validation procedure consists in 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. 1 Introduction Linear Complementarity Problems (LCP) model many important problems in engineering, management and economics. Furthermore linear and quadratic programming problems can be written as LCP. Several algorithms have been developed for solving LCP [11, 21, 22, 25, 26, 31], but few validation methods have been studied to giv...

Three generalizations of the criss-cross method for quadratic programming are presented here. Tucker's, Cottle's and Dantzig's principal pivoting methods are specialized as diagonal and exchange pivots for the linear complementarity problem obtained from a convex quadratic program. A finite criss-cross method, based on least-index resolution, is constructed for solving the LCP. In proving finiteness, orthogonality properties of pivot tableaus and positive semidefiniteness of quadratic matrices are used. In the last section some special cases and two further variants of the quadratic criss-cross method are discussed. If the matrix of the LCP has full rank, then a surprisingly simple algorithm follows, which coincides with Murty's `Bard type schema' in the P matrix case.

A continuous function f with domain X and range f(X) in R n is weakly univalent if there is a sequence of continuous one-to-one functions on X converging to f uniformly on bounded subsets of X . In this article, we establish, under certain conditions, the connectedness of an inverse image f \Gamma1 (q). The univalence results of Radulescu-Radulescu, Mor'e-Rheinboldt, and Gale-Nikaido follow from our main result. We also show that the solution set of a nonlinear complementarity problem corresponding to a continuous P 0 -function is connected if it contains a nonempty bounded clopen set; in particular, the problem will have a unique solution if it has a locally unique solution.

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. We define a minimization problem with simple bounds associated to the horizontal linear complementarity problem (HLCP). When the HLCP is solvable, its solutions are the global minimizers of the associated problem. When the HLCP is feasible, we are able to prove a number of properties of the stationary points of the associated problem. In many cases, the stationary points are solutions of the HLCP. The theoretical results allow us to conjecture that local methods for box constrained optimization applied to the associated problem are efficient tools for solving linear complementarity problems. Numerical experiments seem to confirm this conjecture. Key words. Horizontal linear complementarity problem, linear complementarity problem, bound constrained minimization, optimality conditions, stationary points, global minimizers. AMS (MOS) subject classification. 49M15, 65K05, 90C33. Published in Journal of Global Optimization 6, 1995, pp. 1-15. This work was supported by FAPESP (grants 9...

. We explain and justify a path-following algorithm for solving the equations AC (x) = a, where A is a linear transformation from IR n to IR n , C is a polyhedral convex subset of IR n , and AC is the associated normal map. When AC is coherently oriented, we are able to prove that the path following method terminates at the unique solution of AC (x) = a, which is a generalization of the well known fact that Lemke's method terminates at the unique solution LCP(q; M ) when M is a P--matrix. Otherwise, we identify two classes of matrices which are analogues of the class of copositive--plus and L--matrices in the study of the linear complementarity problem. We then prove that our algorithm processes AC (x) = a when A is the linear transformation associated with such matrices. That is, when applied to such a problem, the algorithm will find a solution unless the problem is infeasible in a well specified sense. 1. Introduction In this paper we are concerned with the Affine Variatio...

. The variational inequality problem is reduced to an optimization problem with a differentiable objective function and simple bounds. Theoretical results are proved, that relate stationary points of the minimization problem to solutions of the variational inequality problem. Perturbations of the original problem are studied and an algorithm that uses the smooth minimization approach for solving monotone problems is defined. Key words. Variational inequalities, box constrained optimization, complementarity. 1 Introduction Let\Omega be a nonempty, closed and convex subset of IR n and F : IR n ! IR n . The finite-dimensional variational inequality problem, denoted by VIP, is to find a vector x 2\Omega such that hF (x); w \Gamma xi 0; for all w 2\Omega : (1) This problem has many interesting applications and its solution using special techniques has been considered extensively in the literature; see, for example, (Ref. 1) and references therein. The linear and nonlinear comp...

This paper applies splitting techniques developed for set-valued maximal monotone operators to monotone affine variational inequalities, including as a special case the classical linear complementarity problem. We give a unified presentation of several splitting algorithms for monotone operators, and then apply these results to obtain two classes of algorithms for affine variational inequalities. The second class resembles classical matrix splitting, but has a novel "underrelaxation " step, and converges under more general conditions. In particular, the convergence proofs do not require the affine operator to be symmetric. We specialize our matrix-splittinglike method to discrete-time optimal control problems formulated as extended linear-quadratic programs in the manner advocated by Rockafellar and Wets. The result is a highly parallel algorithm, which we implement and test on the Connection Machine CM--5 computer family. The affine variational inequality problem is to find a vector x...

A new predictor--corrector interior point algorithm for solving monotone linear complementarity problems (LCP) is proposed, and it is shown to be superlinearly convergent with at least order 1.5, even if the LCP has no strictly complementary solution. Unlike Mizuno's recent algorithm [16], the fast local convergence is attained without any need for estimating the optimal partition. In the special case that a strictly complementary solution does exist, the order of convergence becomes quadratic. The proof relies on an investigation of the asymptotic behavior of first and second order derivatives that are associated with trajectories of weighted centers for LCP. AMS 1991 subject classification: 90C33. Key words. monotone linear complementarity problem, primal-dual interior point method, superlinear convergence, central path. 1 1. Introduction Given n \Theta n real matrices Q and R and a real vector b of order n, the horizontal linear complementarity problem (LCP) is the problem of fin...