A Theorem on Strict Separability of Convex Polyhedra and Its Applications in Optimization

Necessary and sufficient conditions for emptiness of the cones of generalized support vectors

“…It has been previously remarked in [5,19,20] that the problem of linear (strong, strict) separation of two sets A and B may be reduced to the problem of linear (strong or strict, respectively) separation of the origin of R n from the Minkowski difference of these sets. In this subsection, we consider what additional results concerning the separability of the sets A and B can be obtained on the basis of the results gained for the separation of the origin of…”

“…By virtue of convexity and closedness of the sets cl(co(Φ)) −ȳ, cl(co(Ψ )) −x, we obtain that Ω cl(co(Φ))−ȳ , Ω cl(co(Ψ ))−x are convex and compact sets. Owing to (18)- (19), the application of Theorem 3.3 from [20] yields thatx −ȳ ∈ Bd(Ω cl(co(Φ))−ȳ ) andȳ −x ∈ Bd(Ω cl(co(Ψ ))−x ). Due to closedness of Ω cl(co(Φ))−ȳ and Ω cl(co(Ψ ))−x , we getx −ȳ ∈ Ω cl(co(Φ))−ȳ and y−x ∈ Ω cl(co(Ψ ))−x .…”

“…The results of our previous research have been presented in [18][19][20]. A key idea of all above-mentioned papers was to construct the cones of GSVs corresponding to the families of hyperplanes which linearly (strongly or non-strongly) separate the considered sets (convex polyhedra or arbitrary sets of R n ).…”

“…Having solved the optimization program used for minimizing the error-function, based on the optimal value of the objective function, there can be analyzed on degeneracy (or non-degeneracy) the cones of GSVs and GSSVs. However, some of optimization programs may be used for separation of convex polyhedra only in the case when they are disjoint [19,22]. Otherwise, they commonly fail to reveal the degeneracy or non-degeneracy of the cone of GSVs.…”

“…Due to a broad spectrum of practical applications, the theory of the linear separation of sets receives especially much attention [1,5,6,8,[18][19][20]22,23,[25][26][27][28][29][30]. This stipulates the relevance of the degeneracy analysis of the cones of GSVs from both theoretical and practical points of view.…”