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Generalized linear complementarity problems. (English) Zbl 0708.90089

The generalization is twofold. First, the problem is defined for closed convex cones rather than for the non-negative orthant. Second, some, but not all, the results are stated for infinite-dimensional real Hilbert spaces.
Two infinite-dimensional existence results are given. That is, the authors state conditions under which feasibility of a Generalized Linear Complementarity Problem (GLCP) with copositive plus operator implies solvability. Moreover a section is devoted to a finite-dimensional perturbation result.
It is immediately apparent that polyhedral cones have a distinctive role. In the infinite-dimensional setting such cones are defined as finitely generated cones, and hence they live in a finite-dimensional subspace. Despite this, the final result, which characterizes polyhedral cones, is finite-dimensional: if the dimension of the space is finite then “polyhedral cones are the only ones with the property that every copositive plus feasible GLCP is solvable”.
The extension to infinite dimension of this result is one of the open problems that conclude the paper. I will stay in tune to learn, in a forthcoming paper by the same authors, how the mystery resolves.
Reviewer: P.d’Alessandro

MSC:

90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)
90C48 Programming in abstract spaces
52B12 Special polytopes (linear programming, centrally symmetric, etc.)
90C05 Linear programming
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