v2010.10.26 - Convex Optimization

2.13. DUAL CONE & GENERALIZED INEQUALITY 213whereas dual proper polyhedral cone K ∗ has only five:⎧ ⎫ ⎧F1∗ Γ ∗⎪⎨ F2∗ 1 Γ ∗ 2 Γ ∗ 3 Γ ∗ 4⎪⎬ ⎪⎨ Γ ∗G F3∗ 1 Γ ∗ 2 Γ ∗ 6= Γ ∗F4⎪⎩∗ 1 Γ ∗ 4 Γ ∗ 5 Γ ∗ 6Γ ⎪⎭ ⎪⎩∗F5∗ 3 Γ ∗ 4 Γ ∗ 5Γ ∗ 2 Γ ∗ 3 Γ ∗ 5 Γ ∗ 6⎫⎪⎬⎪⎭(485)Six two-dimensional cones, having generators respectively {Γ ∗ 1 Γ ∗ 3} {Γ ∗ 2 Γ ∗ 4}{Γ ∗ 1 Γ ∗ 5} {Γ ∗ 4 Γ ∗ 6} {Γ ∗ 2 Γ ∗ 5} {Γ ∗ 3 Γ ∗ 6} , are relatively interior to dual facets;so cannot be two-dimensional faces of K ∗ (by Definition 2.6.0.0.3).We can check this result (483) by reversing the process; we find6!/((6 −4)! 4!) − 3=12 component simplices in the dual cone. 2.87 Applyingalgorithm (477) to those simplices returns a conically independent set ofgenerators for K equivalent to (482).2.13.11.0.4 Exercise. Reaching proper polyhedral cone interior.Name two extreme directions Γ i of cone K from Example 2.13.11.0.3 whoseconvex hull passes through that cone’s interior. Explain why. Are there twosuch extreme directions of dual cone K ∗ ?2.13.12 coordinates in proper nonsimplicial systemA natural question pertains to whether a theory of unique coordinates,like biorthogonal expansion, is extensible to proper cones whose extremedirections number in excess of ambient spatial dimensionality.2.13.12.0.1 Theorem. Conic coordinates.With respect to vector v in some finite-dimensional Euclidean space R n ,define a coordinate t ⋆ v of point x in full-dimensional pointed closed convexcone Kt ⋆ v(x) sup{t∈ R | x − tv ∈ K} (486)Given points x and y in cone K , if t ⋆ v(x)= t ⋆ v(y) for each and every extremedirection v of K then x = y .⋄2.87 Three combinations of four dual extreme directions are linearly dependent; they belongto the dual facets. But there are no linearly dependent combinations of three dual extremedirections.