v2010.10.26 - Convex Optimization

50 CHAPTER 2. CONVEX GEOMETRY2.1.9.0.2 Corollary. Projection on subspace. 2.10 (1934) [307,3]Orthogonal projection of a convex set on a subspace or nonempty affine setis another convex set.⋄Again, the converse is false. Shadows, for example, are umbral projectionsthat can be convex when the body providing the shade is not.2.2 Vectorized-matrix inner productEuclidean space R n comes equipped with a linear vector inner-product〈y,z〉 y T z (33)We prefer those angle brackets to connote a geometric rather than algebraicperspective; e.g., vector y might represent a hyperplane normal (2.4.2).Two vectors are orthogonal (perpendicular) to one another if and only iftheir inner product vanishes;y ⊥ z ⇔ 〈y,z〉 = 0 (34)When orthogonal vectors each have unit norm, then they are orthonormal.A vector inner-product defines Euclidean norm (vector 2-norm,A.7.1)‖y‖ 2 = ‖y‖ √ y T y , ‖y‖ = 0 ⇔ y = 0 (35)For linear operator A , the adjoint operator A T is defined by [227,3.10]〈y,A T z〉 〈Ay,z〉 (36)For linear operation on a vector, represented by real matrix A , the adjointoperator A T is its transposition.The vector inner-product for matrices is calculated just as it is for vectors;by first transforming a matrix in R p×k to a vector in R pk by concatenatingits columns in the natural order. For lack of a better term, we shall callthat linear bijective (one-to-one and onto [227, App.A1.2]) transformation2.10 For hyperplane representations see2.4.2. For projection of convex sets on hyperplanessee [371,6.6]. A nonempty affine set is called an affine subset (2.3.1.0.1). Orthogonalprojection of points on affine subsets is reviewed inE.4.