v2004.06.19 - Convex Optimization

2.1. CONVEX SET 23sole negative coordinate is their i th (II or IV). Orthant convexity 2.3 is easilyverified by definition (1).2.1.0.8 sum, product, differenceThe vector sum of two convex sets C 1 and C 2C 1 + C 2 = {x + y | x ∈ C 1 , y ∈ C 2 } (7)and Cartesian product{[ xC 1 × C 2 =y]| x ∈ C 1 , y ∈ C 2}(8)remain convex. By additive inverse, we can similarly define the vectordifference of two convex setsC 1 − C 2 = {x − y | x∈ C 1 , y ∈ C 2 } (9)which is convex. Applying this definition to nonempty convex C 1 , the selfdifferenceC 1 − C 1 is generally nonempty, nontrivial, and convex.Convex results are also obtained for scaling κ C , rotation/reflection Q C ,or translation C+α of a convex set C ; all similarly defined.Given any operator T and convex set C , we are prone to write T(C)meaningT(C) ∆ = {T(x) | x∈ C} (10)Given linear operator T , it therefore follows from (7),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 )(11)2.1.0.9.1 Theorem. Intersection. [1, §2.3.1] [30, §2] The intersectionof an arbitrary collection of convex sets is convex.⋄Note that the converse is implicitly false in so far as a convex set can beformed by the intersection of sets that are not.2.3 All orthants are self-dual simplicial cones. (§2.8.3.2, §2.7.3.0.1)