v2007.09.13 - Convex Optimization

184 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS3.1 Convex function3.1.1 real and vector-valued functionVector-valued functionf(X) : R p×k →R M =⎡⎢⎣f 1 (X).f M (X)⎤⎥⎦ (417)assigns each X in its domain domf (a subset of ambient vector space R p×k )to a specific element [188, p.3] of its range (a subset of R M ). Function f(X)is linear in X on its domain if and only if, for each and every Y,Z ∈domfand α , β ∈ Rf(αY + βZ) = αf(Y ) + βf(Z) (418)A vector-valued function f(X) : R p×k →R M is convex in X if and only ifdomf is a convex set and, for each and every Y,Z ∈domf and 0≤µ≤1f(µY + (1 − µ)Z) ≼µf(Y ) + (1 − µ)f(Z) (419)R M +As defined, continuity is implied but not differentiability. Reversing thesense of the inequality flips this definition to concavity. Linear functions are,apparently, simultaneously convex and concave.Vector-valued functions are most often compared (151) as in (419) withrespect to the M-dimensional self-dual nonnegative orthant R M + , a propercone. 3.2 In this case, the test prescribed by (419) is simply a comparisonon R of each entry f i of a vector-valued function f . (2.13.4.2.3) Thevector-valued function case is therefore a straightforward generalization ofconventional convexity theory for a real function. [46,3,4] This conclusionfollows from theory of dual generalized inequalities (2.13.2.0.1) which asserts3.2 The definition of convexity can be broadened to other (not necessarily proper) cones;referred to in the literature as K-convexity. [217]