v2010.10.26 - Convex Optimization

2.13. DUAL CONE & GENERALIZED INEQUALITY 215has no nontrivial nullspace. Because x −t ⋆ v must belong to ∂K by definition,the mapping t ⋆ (x) is equivalent to a convex problem (separable in index i)whose objective (by (330)) is tightly bounded below by 0 :t ⋆ (x) ≡ arg minimizet∈R NN∑i=1Γ ∗Tj(i) (x − t iΓ i )subject to x − t i Γ i ∈ K ,i=1... N(488)where index j ∈ I is dependent on i and where (by (368)) λ = Γj ∗ ∈ R n is anextreme direction of dual cone K ∗ that is normal to a hyperplane supportingK and containing x − t ⋆ iΓ i . Because extreme-direction cardinality N forcone K is not necessarily the same as for dual cone K ∗ , index j must bejudiciously selected from a set I .To prove injectivity when extreme-direction cardinality N > n exceedsspatial dimension, we need only show mapping t ⋆ (x) to be invertible;[128, thm.9.2.3] id est, x is recoverable given t ⋆ (x) :x = arg minimize˜x∈R nN∑i=1Γ ∗Tj(i) (˜x − t⋆ iΓ i )subject to ˜x − t ⋆ iΓ i ∈ K ,i=1... N(489)The feasible set of this nonseparable convex problem is an intersection oftranslated full-dimensional pointed closed convex cones ⋂ i K + t⋆ iΓ i . Theobjective function’s linear part describes movement in normal-direction −Γj∗for each of N hyperplanes. The optimal point of hyperplane intersection isthe unique solution x when {Γj ∗ } comprises n linearly independent normalsthat come from the dual cone and make the objective vanish. Because thedual cone K ∗ is full-dimensional, pointed, closed, and convex by assumption,there exist N extreme directions {Γj ∗ } from K ∗ ⊂ R n that span R n . Sowe need simply choose N spanning dual extreme directions that make theoptimal objective vanish. Because such dual extreme directions preexistby (330), t ⋆ (x) is invertible.Otherwise, in the case N ≤ n , t ⋆ (x) holds coordinates for biorthogonalexpansion. Reconstruction of x is therefore unique.2.13.12.1 reconstruction from conic coordinatesThe foregoing proof of the conic coordinates theorem is not constructive; itestablishes existence of dual extreme directions {Γ ∗ j } that will reconstruct