v2010.10.26 - Convex Optimization

2.1. CONVEX SET 49R nR mR(A T )x px p =A † bA † Ax = x pAx p = bR(A)b{x}0ηN(A)x=x p + ηAx = bAη = 0N(A T )0{b}Figure 16:(confer Figure 161) Action of linear map represented by A∈ R m×n :Component of vector x in nullspace N(A) maps to origin while componentin rowspace R(A T ) maps to range R(A). For any A∈ R m×n , AA † Ax = b(E) and inverse image of b ∈ R(A) is a nonempty affine set: x p + N(A).Inverse image of a convex set F ,f −1 (F) = {X | f(X)∈ F} ⊆ R p×k (32)a single- or many-valued mapping, under any affine function f isconvex.⋄In particular, any affine transformation of an affine set remains affine.[307, p.8] Ellipsoids are invariant to any [sic] affine transformation.Although not precluded, this inverse image theorem does not require auniquely invertible mapping f . Figure 16, for example, mechanizes inverseimage under a general linear map. Example 2.9.1.0.2 and Example 3.5.0.0.2offer further applications.Each converse of this two-part theorem is generally false; id est, givenf affine, a convex image f(C) does not imply that set C is convex, andneither does a convex inverse image f −1 (F) imply set F is convex. Acounterexample, invalidating a converse, is easy to visualize when the affinefunction is an orthogonal projector [331] [250]: