v2007.09.13 - Convex Optimization

2.13. DUAL CONE & GENERALIZED INEQUALITY 157Now consider the (closed convex) dual cone:K ∗ = {y | 〈A svec X , y〉 ≥ 0 for all X ≽ 0} ⊆ R m= { y | 〈svec X , A T y〉 ≥ 0 for all X ≽ 0 }= { y | svec −1 (A T y) ≽ 0 } (329)that follows from (322) and leads to an equally peculiar halfspace-descriptionK ∗ = {y ∈ R m |m∑y j A j ≽ 0} (330)j=1The summation inequality with respect to the positive semidefinite cone isknown as a linear matrix inequality. [44] [101] [191] [270]When the A j matrices are linearly independent, function g(y) ∆ = ∑ y j A jon R m is a linear bijection. The inverse image of the positive semidefinitecone under g(y) must therefore have dimension m . In that circumstance,the dual cone interior is nonemptyint K ∗ = {y ∈ R m |m∑y j A j ≻ 0} (331)j=1having boundary∂K ∗ = {y ∈ R m |m∑y j A j ≽ 0,j=1m∑y j A j ⊁ 0} (332)j=12.13.6 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 in X ;X = [ Γ 1 Γ 2 · · · Γ N ] ∈ R n×N (240)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: