v2010.10.26 - Convex Optimization
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v2010.10.26 - Convex Optimization

v2010.10.26 - Convex Optimization v2010.10.26 - Convex Optimization

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186 CHAPTER 2. CONVEX GEOMETRY2.13.6.1 Facet normal & extreme directionWe see from (363) that the conically independent generators of cone K(namely, the extreme directions of pointed closed convex cone K constitutingthe N columns of X) each define an inward-normal to a hyperplanesupporting dual cone K ∗ (2.4.2.6.1) and exposing a dual facet when N isfinite. Were K ∗ pointed and finitely generated, then by closure the conjugatestatement would also hold; id est, the extreme directions of pointed K ∗ eachdefine an inward-normal to a hyperplane supporting K and exposing a facetwhen N is finite. Examine Figure 56 or Figure 62, for example.We may conclude, the extreme directions of proper polyhedral K arerespectively orthogonal to the facets of K ∗ ; likewise, the extreme directionsof proper polyhedral K ∗ are respectively orthogonal to the facets of K .2.13.7 Biorthogonal expansion by example2.13.7.0.1 Example. Relationship to dual polyhedral cone.Simplicial cone K illustrated in Figure 63 induces a partial order on R 2 . Allpoints greater than x with respect to K , for example, are contained in thetranslated cone x + K . The extreme directions Γ 1 and Γ 2 of K do not makean orthogonal set; neither do extreme directions Γ 3 and Γ 4 of dual cone K ∗ ;rather, we have the biorthogonality condition, [365]Γ T 4 Γ 1 = Γ T 3 Γ 2 = 0Γ T 3 Γ 1 ≠ 0, Γ T 4 Γ 2 ≠ 0(394)Biorthogonal expansion of x ∈ K is thenx = Γ 1Γ T 3 xΓ T 3 Γ 1+ Γ 2Γ T 4 xΓ T 4 Γ 2(395)where Γ T 3 x/(Γ T 3 Γ 1 ) is the nonnegative coefficient of nonorthogonal projection(E.6.1) of x on Γ 1 in the direction orthogonal to Γ 3 (y in Figure 163, p.717),and where Γ T 4 x/(Γ T 4 Γ 2 ) is the nonnegative coefficient of nonorthogonalprojection of x on Γ 2 in the direction orthogonal to Γ 4 (z in Figure 163);they are coordinates in this nonorthogonal system. Those coefficients mustbe nonnegative x ≽ K0 because x ∈ K (325) and K is simplicial.If we ascribe the extreme directions of K to the columns of a matrixX [ Γ 1 Γ 2 ] (396)

2.13. DUAL CONE & GENERALIZED INEQUALITY 187Γ 4Γ 2Γ 3K ∗xzK yΓ 10Γ 1 ⊥ Γ 4Γ 2 ⊥ Γ 3K ∗w − KuwFigure 63: (confer Figure 163) Simplicial cone K in R 2 and its dual K ∗ drawntruncated. Conically independent generators Γ 1 and Γ 2 constitute extremedirections of K while Γ 3 and Γ 4 constitute extreme directions of K ∗ . Dottedray-pairs bound translated cones K . Point x is comparable to point z(and vice versa) but not to y ; z ≽ Kx ⇔ z − x ∈ K ⇔ z − x ≽ K0 iff ∃nonnegative coordinates for biorthogonal expansion of z − x . Point y is notcomparable to z because z does not belong to y ± K . Translating a negatedcone is quite helpful for visualization: u ≼ Kw ⇔ u ∈ w − K ⇔ u − w ≼ K0.Points need not belong to K to be comparable; e.g., all points less than w(with respect to K) belong to w − K .

2.13. DUAL CONE & GENERALIZED INEQUALITY 187Γ 4Γ 2Γ 3K ∗xzK yΓ 10Γ 1 ⊥ Γ 4Γ 2 ⊥ Γ 3K ∗w − KuwFigure 63: (confer Figure 163) Simplicial cone K in R 2 and its dual K ∗ drawntruncated. Conically independent generators Γ 1 and Γ 2 constitute extremedirections of K while Γ 3 and Γ 4 constitute extreme directions of K ∗ . Dottedray-pairs bound translated cones K . Point x is comparable to point z(and vice versa) but not to y ; z ≽ Kx ⇔ z − x ∈ K ⇔ z − x ≽ K0 iff ∃nonnegative coordinates for biorthogonal expansion of z − x . Point y is notcomparable to z because z does not belong to y ± K . Translating a negatedcone is quite helpful for visualization: u ≼ Kw ⇔ u ∈ w − K ⇔ u − w ≼ K0.Points need not belong to K to be comparable; e.g., all points less than w(with respect to K) belong to w − K .

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