This paper presents a connection between qualitative matrix theory and linear complementarity problems (LCPs). An LCP is said to be sign-solvable if the set of the sign patterns of the solutions is uniquely determined by the sign patterns of the given coefficients. We provide a characterization for sign-solvable LCPs such that the coefficient matrix has nonzero diagonals, which can be tested in...

We describe a “condition” number for the linear complementarity problem (LCP), which characterizes the degree of difficulty for its solution when a potential reduction algorithm is used. Consequently, we develop a class of LCPs solvable in polynomial time. The result suggests that the convexity (or positive semidefiniteness) of the LCP may not be the basic issue that separates LCPs solvable and...

We present an algebraic version of an iterative multigrid method for obstacle problems, called projected algebraic multigrid (PAMG) here. We show that classical algebraic multigrid algorithms can easily be extended to deal with this kind of problem. This paves the way for efficient multigrid solution of obstacle problems with partial differential equations arising, for example, in financial eng...

We report and analyze the results of our computational testing of branchand-cut for the complementarity-constrained optimization problem (CCOP). Besides the MIP cuts commonly present in commercial optimization software, we used inequalities that explore complementarity constraints. To do so, we generalized two families of cuts proposed earlier by de Farias, Johnson, and Nemhauser that had never...

‎A full Nesterov-Todd (NT) step infeasible interior-point algorithm‎ ‎is proposed for solving monotone linear complementarity problems‎ ‎over symmetric cones by using Euclidean Jordan algebra‎. ‎Two types of‎ ‎full NT-steps are used‎, ‎feasibility steps and centering steps‎. ‎The‎ ‎algorithm starts from strictly feasible iterates of a perturbed‎ ‎problem‎, ‎and, using the central path and feasi...

In this report an iterative method from the theory of maximal monotone operators is transfered into the context of linear complementarity problems and numerical tests are performed on contact problems from the field of rigid multibody dynamics.

We present a new interior-point potential-reduction algorithm for solving monotone linear complementarity problems (LCPs) that have a particular special structure: their matrix M ∈ Rn×n can be decomposed as M = ΦU +Π0, where the rank of Φ is k < n, and Π0 denotes Euclidean projection onto the nullspace of Φ⊤. We call such LCPs projective. Our algorithm solves a monotone projective LCP to relati...

Recently, the globally uniquely solvable (GUS) property of the linear transformation M ∈ Rn×n in the second-order cone linear complementarity problem (SOCLCP) receives much attention and has been studied substantially. Yang and Yuan [30] contributed a new characterization of the GUS property of the linear transformation, which is formulated by basic linear-algebra-related properties. In this pa...

We define a new fundamental constant associated with a P-matrix and show that this constant has various useful properties for the P-matrix linear complementarity problems (LCP). In particular, this constant is sharper than the Mathias-Pang constant in deriving perturbation bounds for the P-matrix LCP. Moreover, this new constant defines a measure of sensitivity of the solution of the P-matrix L...