v2010.10.26 - Convex Optimization

2.1. CONVEX SET 47By additive inverse, we can similarly define vector difference of two convexsetsC 1 − C 2 = {x − y | x ∈ C 1 , y ∈ C 2 } (27)which is convex. Applying this definition to nonempty convex set C 1 , itsself-difference C 1 − C 1 is generally nonempty, nontrivial, and convex; e.g.,for any convex cone K , (2.7.2) the set K − K constitutes its affine hull.[307, p.15]Cartesian product of convex sets{[ ]} [ ]x C1C 1 × C 2 = | x ∈ Cy 1 , y ∈ C 2 =(28)C 2remains convex. The converse also holds; id est, a Cartesian product isconvex iff each set is. [199, p.23]Convex results are also obtained for scaling κ C of a convex set C ,rotation/reflection Q C , or translation C+ α ; each similarly defined.Given any operator T and convex set C , we are prone to write T(C)meaningT(C) {T(x) | x∈ C} (29)Given linear operator T , it therefore follows from (26),T(C 1 + C 2 ) = {T(x + y) | x∈ C 1 , y ∈ C 2 }= {T(x) + T(y) | x∈ C 1 , y ∈ C 2 }= T(C 1 ) + T(C 2 )(30)2.1.9 inverse imageWhile epigraph (3.5) of a convex function must be convex, it generallyholds that inverse image (Figure 15) of a convex function is not. The mostprominent examples to the contrary are affine functions (3.4):2.1.9.0.1 Theorem. Inverse image. [307,3]Let f be a mapping from R p×k to R m×n .The image of a convex set C under any affine functionis convex.f(C) = {f(X) | X ∈ C} ⊆ R m×n (31)