Richard W. Cottle

The linear complementarity problem

An article holding receptacle, such as an expandable envelope, is releasably secured to the steering column of an automotive vehicle by means of an elasticized band wrapped partly around the steering column hooked at both ends to a clip from which the envelope is releasably attached. The length of the band is such that, when engaged at its ends with the clip, it is under sufficient tension to be retained snugly against the steering column. The envelope is secured to the clip by releasable means such as a paper fastener and includes means to prevent skewing of the envelope about the fastener.

Complementary pivot theory of mathematical programming

Abstract : Problems of the form: Find w and z satisfying w = q + Mz, w = or 0, z = or 0, zw = 0 play a fundamental role in mathematical programming. This paper describes the role of such problems in linear programming, quadratic programming and bimatrix game theory and reviews the computational procedures of Lemke and Howson, Lemke, and Dantzig and Cottle.

Manifestations of the Schur complement

In this paper the author is concerned with some of the ways in which the Schur complement can be used in numerical linear algebra.

A generalization of the linear complementarity problem

Abstract The linear complementarity problem: find z∈Rp satisfying w = q + M z w ⩾ 0 , z ⩾ 0 ( LCP ) z T w = 0 is generalized to a problem in which the matrix M is not square. A solution technique similar to C. E. Lemkes (1965) method for solving (LCP) is given. The method is discussed from a graph-theoretic viewpoint and closely parallels a proof of Sperners lemma by D. I. A. Cohen (1967) and some work of H. Scarf (1967) on approximating fixed points of a continuous mapping of a simplex into itself.

Sufficient matrices and the linear complementarity problem

Abstract We pose and answer two questions about solutions of the linear complementarity problem (LCP). The first question is concerned with the conditions on a matrix M which guarantee that for every vector q , the solutions of the LCP ( q , M ) are identical to the Karush-Kuhn-Tucker points of the natural quadratic program associated with ( q , M ). In answering this question we introduce the class of “row sufficient” matrices. The transpose of such a matrix is what we call “column sufficient”. The latter matrices turn out to furnish the answer to our second question, which asks for the conditions on M under which the solution set of ( q , M ) is convex for every q . In addition to these two main results, we discuss the connections of these twonew matrix classes with other well-known matrix classes in linear complementarity theory.

Polyhedral sets having a least element

For a fixedm × n matrixA, we consider the family of polyhedral setsXb ={x|Ax ≥ b}, b ∈ Rm, and prove a theorem characterizing, in terms ofA, the circumstances under which every nonemptyXb has a least element. In the special case whereA contains all the rows of ann × n identity matrix, the conditions are equivalent toAT being Leontief. Among the corollaries of our theorem, we show the linear complementarity problem always has a unique solution which is at the same time a least element of the corresponding polyhedron if and only if its matrix is square, Leontief, and has positive diagonals.

Monotone solutions of the parametric linear complementarity problem

The parametric linear complementarity problem under study here is given by the conditionsq + αp + Mz ≥ 0,α ≥ 0,z ≥ 0,zT(q + αp + Mz) = 0 whereq is nonnegative. This paper answers three questions including the following one raised by G. Maier: What are the necessary and sufficient conditions onM guaranteeing that for every nonnegative starting pointq and every directionp the components of the solution to the parametric linear complementarity problem are nondecreasing functions of the parameterα?

On classes of copositive matrices

Abstract Characterizations are given of copositive, strictly copositive, and copositive plus matrices, and their quadratic forms, together with relationships of these with positive semidefinite matrices.

Pseudo-monotone complementarity problems in Hilbert space

In this paper, some existence results for a nonlinear complementarity problem involving a pseudo-monotone mapping over an arbitrary closed convex cone in a real Hilbert space are established. In particular, some known existence results for a nonlinear complementarity problem in a finite-dimensional Hilbert space are generalized to an infinite-dimensional real Hilbert space. Applications to a class of nonlinear complementarity problems and the study of the post-critical equilibrium state of a thin elastic plate subjected to unilateral conditions are given.

On the uniqueness of solutions to linear complementarity problems

This paper characterizes the classU of all realn×n matricesM for which the linear complementarity problem (q, M) has a unique solution for all realn-vectorsq interior to the coneK(M) of vectors for which (q, M) has any solution at all. It is shown that restricting the uniqueness property to the interior ofK(M) is necessary because whenU, the problem (q, M) has infinitely many solutions ifq belongs to the boundary of intK(M). It is shown thatM must have nonnegative principal minors whenU andK(M) is convex. Finally, it is shown that whenM has nonnegative principal minors, only one of which is 0, andK(M)≠Rn, thenU andK(M) is a closed half-space.