Some geometrical aspects of semidefinite linear complementarity problems

This article studies some geometrical aspects of the semidefinite linear complementarity problem (SDLCP), which can be viewed as a generalization of the well-known linear complementarity problem (LCP). SDLCP is a special case of a complementarity problem over a closed convex cone, where the cone considered is the closed convex cone of positive semidefinite matrices. It arises naturally in the unified formulation of a pair of primal-dual semidefinite programming problems. In this article, we introduce the notion of complementary cones in the semidefinite setting using the faces of the cone of positive semidefinite matrices and show that unlike complementary cones induced by an LCP, semidefinite complementary cones need not be closed. However, under R ₀-property of the linear transformation, closedness of all the semidefinite complementary cones induced by L is ensured. We also introduce the notion of a principal subtransformation with respect to a face of the cone of positive semidefinite matrices and show that for a self-adjoint linear transformation, strict copositivity is equivalent to strict semimonotonicity of each principal subtransformation. Besides the above, various other solution properties of SDLCP will be interpreted and studied geometrically.     

Errata: On semidefinite linear complementarity problems

We correct an error in the statement of Theorem 8 in [1]. Received: January 3, 2001 / Accepted: February 26, 2001?Published online May 18, 2001     

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.     

A comparison of error bounds for linear complementarity problems of H-matrices

We give new error bounds for the linear complementarity problem when the involved matrix is an H-matrix with positive diagonals. We find classes of H-matrices for which the new bounds improve considerably other previous bounds. We also show advantages of these new bounds with respect the computational cost. A new perturbation bound of H-matrices linear complementarity problems is also presented.     

Spectral analysis of coupled linear complementarity problems

This note deals with the so-called cone-constrained bivariate eigenvalue problem. The equilibrium model under consideration is a system of linear complementarity problems

     

On semidefinite linear complementarity problems

Given a linear transformation L:? ⁿ →? ⁿ and a matrix Q∈? ⁿ , where ? ⁿ is the space of all symmetric real n×n matrices, we consider the semidefinite linear complementarity problem SDLCP(L,? ⁿ ₊,Q) over the cone ? ⁿ ₊ of symmetric n×n positive semidefinite matrices. For such problems, we introduce the P-property and its variants, Q- and GUS-properties. For a matrix A∈R ^n×n , we consider the linear transformation L _A :? ⁿ →? ⁿ defined by L _A (X):=AX+XA ^T and show that the P- and Q-properties for L _A are equivalent to A being positive stable, i.e., real parts of eigenvalues of A are positive. As a special case of this equivalence, we deduce a theorem of Lyapunov. Received: March 1999 / Accepted: November 1999?Published online April 20, 2000     

Global s-type error bound for the extended linear complementarity problem and applications

For the extended linear complementarity problem over an affine subspace, we first study some characterizations of (strong) column/row monotonicity and (strong) R ₀-property. We then establish global s-type error bound for this problem with the column monotonicity or R ₀-property, especially for the one with the nondegeneracy and column monotonicity, and give several equivalent formulations of such error bound without the square root term for monotone affine variational inequality. Finally, we use this error bound to derive some properties of the iterative sequence produced by smoothing methods for solving such a problem under suitable assumptions. Received: May 2, 1999 / Accepted: February 21, 2000?Published online July 20, 2000     

On cone orderings and the linear complementarity problem

This paper first generalizes a characterization of polyhedral sets having least elements, which is obtained by Cottle and Veinott [6], to the situation in which Euclidean space is partially ordered by some general cone ordering (rather than the usual ordering). We then use this generalization to establish the following characterization of the class $\text{C}$ of matrices ($\text{C}$ arises as a generalization of the class of Z-matrices; see [4], [13], [14]): M∈$\text{C}$ if and only if for every vector q for which the linear complementarity problem (q,M) is feasible, the problem (q,M) has a solution which is the least element of the feasible set of (q,M) with respect to a cone ordering induced by some simplicial cone. This latter result generalizes the characterizations of K-and Z-matrices obtained by Cottle and Veinott [6] and Tamir [21], respectively.     

Solution of linear complementarity problems using minimization with simple bounds

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.This work was supported by FAPESP (grants 90-3724-6 and 91-2441-3), CNPq and FAEP (UNICAMP).     

Numerical validation of solutions of linear complementarity problems

Summary. This paper proposes a validation method for solutions of linear complementarity problems. The validation procedure consists of 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. Received August 21, 1997 / Revised version July 2, 1998     

On semidefinite representations of non-closed sets

Spectrahedra are sets defined by linear matrix inequalities. Projections of spectrahedra are called semidefinitely representable sets. Both kinds of sets are of practical use in polynomial optimization, since they occur as feasible sets in semidefinite programming. There are several recent results on the question which sets are semidefinitely representable. So far, all results focus on the case of closed sets.In this work we develop a new method to prove semidefinite representability of sets which are not closed. For example, the interior of a semidefinitely representable set is shown to be semidefinitely representable. More general, one can remove faces of a semidefinitely representable set and preserve semidefinite representability, as long as the faces are parametrized in a suitable way.     

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.     

A non-interior continuation method for generalized linear complementarity problems

In this paper, we propose a non-interior continuation method for solving generalized linear complementarity problems (GLCP) introduced by Cottle and Dantzig. The method is based on a smoothing function derived from the exponential penalty function first introduced by Kort and Bertsekas for constrained minimization. This smoothing function can also be viewed as a natural extension of Chen-Mangasarian’s neural network smooth function. By using the smoothing function, we approximate GLCP as a family of parameterized smooth equations. An algorithm is presented to follow the smoothing path. Under suitable assumptions, it is shown that the algorithm is globally convergent and local Q-quadratically convergent. Few preliminary numerical results are also reported. Received September 3, 1997 / Revised version received April 27, 1999?Published online July 19, 1999     

Polynomiality of primal-dual algorithms for semidefinite linear complementarity problems based on the Kojima-Shindoh-Hara family of directions

Received August 29, 1996 / Revised version received May 1, 1998 Published online October 21, 1998     

A note on the implementation of the successive overrelaxation method for linear complementarity problems

For the iterative solution of linear complementary problems often the (projected) successive overrelaxation (SOR) method is discussed. Using a columnwise procedure an implementation of the SOR method is proposed which has advantages for the implementation on vector and parallel computers. In addition this procedure saves computational effort whenever a component of the iteration vector is set to zero.     

Farkas' Lemma, other theorems of the alternative, and linear programming in infinite-dimensional spaces: a purely linear-algebraic approach

We briefly consider several formulations of Farkas' Lemma first. Then we assume the setting of two vector spaces, one of them being linearly ordered, over a linearly ordered field till the end of this article. In this setting, we state a generalized version of Farkas' Lemma and prove it in a purely linear-algebraic way. Afterwards, we present Theorems of Motzkin, Tucker, Carver, Dax, and some other theorems of the alternative that characterize consistency of a finite system of linear inequalities. We also mention the Key Theorem, which is a related result. Finally, we use Farkas' Lemma to prove the Duality Theorem for linear programming (with a finite number of linear constraints). The Duality Theorem that is proved here covers, among others, linear programming in a real vector space of finite or infinite dimension and lexicographic linear programming.     

Uniform bounds on the 1-norm of the inverse of lower triangular Toeplitz matrices

A uniform bound on the 1-norm is given for the inverse of a lower triangular Toeplitz matrix with non-negative monotonically decreasing entries whose limit is zero. The new bound is sharp under certain specified constraints. This result is then employed to throw light upon a long standing open problem posed by Brunner concerning the convergence of the one-point collocation method for the Abel equation. In addition, the recent conjecture of Gauthier et al. is proved.     

Q-superlinear convergence of the iterates in primal-dual interior-point methods

Sufficient conditions are given for the Q-superlinear convergence of the iterates produced by primal-dual interior-point methods for linear complementarity problems. It is shown that those conditions are satisfied by several well known interior-point methods. In particular it is shown that the iteration sequences produced by the simplified predictor–corrector method of Gonzaga and Tapia, the simplified largest step method of Gonzaga and Bonnans, the LPF+ algorithm of Wright, the higher order methods of Wright and Zhang, Potra and Sheng, and Stoer, Wechs and Mizuno are Q-superlinearly convergent. Received: February 9, 2000 / Accepted: February 20, 2001?Published online May 3, 2001