v2007.11.26 - Convex Optimization

2.13. DUAL CONE & GENERALIZED INEQUALITY 157semidefinite cone and R(A). In that case, convex cone K has relativeinteriorrel int K = {A svec X | X ≻ 0} (328)and boundaryrel ∂K = {A svec X | X ≽ 0, X ⊁ 0} (329)Now consider the (closed convex) dual cone:K ∗ = {y | 〈z , y〉 ≥ 0 for all z ∈ K} ⊆ R m= {y | 〈z , y〉 ≥ 0 for all z = A svec X , X ≽ 0}= {y | 〈A svec X , y〉 ≥ 0 for all X ≽ 0}= { y | 〈svec X , A T y〉 ≥ 0 for all X ≽ 0 }= { y | svec −1 (A T y) ≽ 0 } (330)that follows from (323) and leads to an equally peculiar halfspace-descriptionK ∗ = {y ∈ R m |m∑y j A j ≽ 0} (331)j=1The summation inequality with respect to the positive semidefinite cone isknown as a linear matrix inequality. [44] [102] [193] [274]When the A j matrices are linearly independent, function g(y) ∆ = ∑ y j A jon R m is a linear bijection. Inverse image of the positive semidefinite coneunder g(y) must therefore have dimension m . In that circumstance, thedual cone interior is nonemptyint K ∗ = {y ∈ R m |m∑y j A j ≻ 0} (332)j=1having boundary∂K ∗ = {y ∈ R m |m∑y j A j ≽ 0,j=1m∑y j A j ⊁ 0} (333)j=1