v2010.10.26 - Convex Optimization

220 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)⎤⎥⎦ (495)assigns each X in its domain domf (a subset of ambient vector space R p×k )to a specific element [258, 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) (496)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) (497)R M +As defined, continuity is implied but not differentiability (nor smoothness). 3.2Apparently some, but not all, nonlinear functions are convex. Reversing senseof the inequality flips this definition to concavity. Linear (and affine3.4) 3.3functions attain equality in this definition. Linear functions are thereforesimultaneously convex and concave.Vector-valued functions are most often compared (182) as in (497) withrespect to the M-dimensional selfdual nonnegative orthant R M + , a propercone. 3.4 In this case, the test prescribed by (497) 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. This conclusion followsfrom theory of dual generalized inequalities (2.13.2.0.1) which assertsf convex w.r.t R M + ⇔ w T f convex ∀w ∈ G(R M∗+ ) (498)3.2 Figure 68b illustrates a nondifferentiable convex function. Differentiability is certainlynot a requirement for optimization of convex functions by numerical methods; e.g., [243].3.3 While linear functions are not invariant to translation (offset), convex functions are.3.4 Definition of convexity can be broadened to other (not necessarily proper) cones.Referred to in the literature as K-convexity, [294] R M∗+ (498) generalizes to K ∗ .