v2010.10.26 - Convex Optimization

260 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS3.6.3 first-order convexity condition, vector-valued fNow consider the first-order necessary and sufficient condition for convexity ofa vector-valued function: Differentiable function f(X) : R p×k →R M is convexif and only if domf is open, convex, and for each and every X,Y ∈ domff(Y ) ≽R M +f(X) +→Y −Xdf(X) f(X) + d dt∣ f(X+ t (Y − X)) (614)t=0→Y −Xwhere df(X) is the directional derivative 3.21 [219] [332] of f at X in directionY −X . This, of course, follows from the real-valued function case: by dualgeneralized inequalities (2.13.2.0.1),f(Y ) − f(X) −where→Y −Xdf(X) ≽R M +→Y −Xdf(X) =0 ⇔⎡⎢⎣〈〉→Y −Xf(Y ) − f(X) − df(X) , w ≥ 0 ∀w ≽ 0R M∗+(615)tr ( ∇f 1 (X) T (Y − X) ) ⎤tr ( ∇f 2 (X) T (Y − X) )⎥.tr ( ∇f M (X) T (Y − X) ) ⎦ ∈ RM (616)Necessary and sufficient discretization (498) allows relaxation of thesemiinfinite number of conditions {w ≽ 0} instead to {w ∈ {e i , i=1... M }}the extreme directions of the selfdual nonnegative orthant. Each extreme→Y −Xdirection picks out a real entry f i and df(X) ifrom the vector-valuedfunction and its directional derivative, then Theorem 3.6.2.0.1 applies.The vector-valued function case (614) is therefore a straightforwardapplication of the first-order convexity condition for real functions to eachentry of the vector-valued function.3.21 We extend the traditional definition of directional derivative inD.1.4 so that directionmay be indicated by a vector or a matrix, thereby broadening scope of the Taylor series(D.1.7). The right-hand side of inequality (614) is the first-order Taylor series expansionof f about X .