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Preprints, Working Papers, ... Year : 2024

On Covering Euclidean Spaces with Q-arrangements of Cones

Abstract

This paper is concerned with a covering problem of R n by stitching 2 n cones along their common facets. The cones could be degenerate and are allowed to overlap. This covering problem provides a geometric equivalent reformulation of a relatively old, yet unsolved, problem that originated in the optimization community: under which conditions on the n × n matrix M , does the so called linear complementarity problem given by w-M z = q, w, z ≥ 0, and w.z = 0, have a solution (w, z) for all vectors q ∈ R n ? If a solution exists for all q, the matrix M is called a Q-matrix. We investigate the regions where no solution exists (so called holes) and show that they only occur around specific vectors for n ≤ 3. This property allows to fully characterize Q-matrices for n ≤ 3 as it shows that the covering reduces to studying locally the problem. In particular, Theorems 15 generalizes [1, Theorem 4.7] as it drops the strong non-degeneracy requirement. This in turn provides a symbolic characterization for Q-matrices for n ≤ 3, that is an automatically generated list of constraints on the entries of the matrix that are satisfied if and only if the matrix is a Q-matrix.
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hal-04444572 , version 1 (08-02-2024)

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  • HAL Id : hal-04444572 , version 1

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Khalil Ghorbal, Christelle Kozaily. On Covering Euclidean Spaces with Q-arrangements of Cones. 2024. ⟨hal-04444572⟩
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