Abstract
An interior point of a triangle is calledCP-point if its orthogonal projection on the line containing each side lies in the relative interior of that side. In classical mathematics, interest in the concept of regularity of a triangle is mainly centered on the property of every interior point of the triangle being a CP-point. We generalize the concept of regularity using this property, and extend this work to simplicial cones in ℝn, and derive necessary and sufficient conditions for this property to hold in them. These conditions highlight the geometric properties of Z-matrices. We show that these concepts have important ramifications in algorithmic studies of the linear complementarity problem. We relate our results to other well known properties of square matrices.
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This paper is dedicated to the memory of Paolo M. Camerini of Politecnico di Milano whose untimely passing is a great loss to his family and friends and to the optimization profession.
Partially supported by NSF grants ECS-8521183 and ECS-8704052 and by NATO grant RG85-0240.
Supported in part by Control Data Corporation grant 84V101 and AFOSR grant 85-0250.
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Kelly, L.M., Murty, K.G. & Watson, L.T. CP-rays in simplicial cones. Mathematical Programming 48, 387–414 (1990). https://doi.org/10.1007/BF01582265
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DOI: https://doi.org/10.1007/BF01582265