v2004.06.19 - Convex Optimization

114 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS3.1 Convex function3.1.1 Vector-valued functionThe vector-valued continuous function f(X) : R p×k →R M is convex in X ifand only if domf is a convex set and, for each and every Y,Z ∈domf andall 0≤µ≤1,f(µY + (1 − µ)Z) ≼µf(Y ) + (1 − µ)f(Z) (276)R M +Reversing the sense of the inequality flips this definition to concavity.Since comparison of vectors here is with respect to R M + (196), theM-dimensional nonnegative orthant, the test prescribed by (276) is simplya comparison on R of each entry of the vector function. The vector-valuedfunction case is therefore a straightforward generalization of conventionalconvexity theory for a real function. [1, §3, §4]This same conclusion also follows from theory of generalized inequality(§2.8.2.0.1) that impliesf convex ⇔ w T f convex ∀w ≽ 0 (277)shown by substituting the defining inequality (276). Discretization(§2.8.2.1.3) allows relaxation of the semi-infinite number of constraints w ≽ 0to: for each and every w ∈ {e i , i=1... M} (the standard basis for R M anda minimal set of generators (§2.6.4.1) for R M + ) from which the stated conclusionfollows.Relation (277) further implies the space of all vector-valued convex functionsis a closed convex cone. Indeed, any nonnegatively weighted sum ofconvex functions remains convex. Certainly any nonnegative sum of realconvex functions remains convex.When f(X) instead satisfies, for each and every distinct Y and Z indomf and all 0