v2010.10.26 - Convex Optimization

184 CHAPTER 2. CONVEX GEOMETRYthat follows from (378) and leads to an equally peculiar halfspace-descriptionK ∗ = {y ∈ R m |m∑y j A j ≽ 0} (387)j=1The summation inequality with respect to positive semidefinite cone S n + isknown as linear matrix inequality. [59] [151] [262] [362] Although we alreadyknow that the dual cone is convex (2.13.1), inverse image theorem 2.1.9.0.1certifies convexity of K ∗ which is the inverse image of positive semidefinitecone S n + under linear transformation g(y) ∑ y j A j . And although wealready know that the dual cone is closed, it is certified by (382). By theinverse image closedness theorem, dual cone relative interior may always beexpressedm∑rel int K ∗ = {y ∈ R m | y j A j ≻ 0} (388)Function g(y) on R m is an isomorphism when the vectorized A j matricesare linearly independent; hence, uniquely invertible. Inverse image K ∗ musttherefore have dimension equal to dim ( R(A T )∩ svec S n +)(49) and relativeboundaryrel ∂K ∗ = {y ∈ R m |j=1m∑y j A j ≽ 0,j=1m∑y j A j ⊁ 0} (389)When this dimension equals m , then dual cone K ∗ is full-dimensionalj=1rel int K ∗ = int K ∗ (14)which implies: closure of convex cone K is pointed (310).2.13.6 Dual of pointed polyhedral coneIn a subspace of R n , now we consider a pointed polyhedral cone K given interms of its extreme directions Γ i arranged columnar inX = [ Γ 1 Γ 2 · · · Γ N ] ∈ R n×N (280)The extremes theorem (2.8.1.1.1) provides the vertex-description of apointed polyhedral cone in terms of its finite number of extreme directionsand its lone vertex at the origin: