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The purpose of the paper is to establish the conditions which are necessary and sufficient for the cones of generalized (strong, strict) support vectors of a set in a finite-dimensional Euclidean space to be empty. In the present paper, an application of the proposed in J Optim Theory Appl (Gabidullina, J. Optim. Theory Appl. 158(1), 145-171, 2013) linear separability criterion for verification on emptiness or nonemptiness of the cones of GSVs (generalized support vectors) is also studied. We carry out the complete degeneracy analysis of the cones of GSVs for the different kinds of nonempty sets of Euclidean space. We present the different applications of the degeneracy analysis of the cones of GSVs as well.

Abstract. This paper is aimed at presenting a systematic survey of the existing now different formulations for the problem of projection of the origin of the Euclidean space onto the convex polyhedron (PPOCP). In the present paper, there are investigated the reduction of the projection program to the problems of quadratic programming, maximin, linear complementarity, and nonnegative least squares. Such reduction justifies the opportunity of utilizing a much more broad spectrum of powerful tools of mathematical programming for solving PPOCP. The paper's goal is to draw the attention of a wide range of research at the different formulations of the projection problem, which remain largely unknown due to the fact that the papers (addressing the subject of concern) are published even though on the adjacent, but other topics, or only in the conference proceedings.Keywords: projection, convex polyhedron, quadratic programming, maximin problem, complementarity problem, nonnegative least squares problem MSC 2010: 90C30, 90C25, 90C20, 90C33, 65K05

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