v2007.09.13 - Convex Optimization

2.13. DUAL CONE & GENERALIZED INEQUALITY 159We may conclude the extreme directions of polyhedral proper K arerespectively orthogonal to the facets of K ∗ ; likewise, the extreme directionsof polyhedral proper 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 49 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 notmake an orthogonal set; neither do extreme directions Γ 3 and Γ 4 of dualcone K ∗ ; rather, we have the biorthogonality condition, [273]Γ T 4 Γ 1 = Γ T 3 Γ 2 = 0Γ T 3 Γ 1 ≠ 0, Γ T 4 Γ 2 ≠ 0(335)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(336)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 , and whereΓ T 4 x/(Γ T 4 Γ 2 ) is the nonnegative coefficient of nonorthogonal projection ofx on Γ 2 in the direction orthogonal to Γ 4 ; they are coordinates in thisnonorthogonal system. Those coefficients must be nonnegative x ≽ K0because x ∈ K (281) and K is simplicial.If we ascribe the extreme directions of K to the columns of a matrixX ∆ = [ Γ 1 Γ 2 ] (337)then we find that the pseudoinverse transpose matrix[]X †T 1 1= Γ 3 ΓΓ T 43 Γ 1 Γ T 4 Γ 2(338)holds the extreme directions of the dual cone. Therefore,x = XX † x (344)