v2007.11.26 - Convex Optimization

214 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS3.1.10 first-order convexity condition, vector functionNow consider the first-order necessary and sufficient condition for convexityof a vector-valued function: Differentiable function f(X) : R p×k →R M isconvex if and only if domf is open, convex, and for each and everyX,Y ∈ domff(Y ) ≽R M +f(X) +→Y −Xdf(X)= f(X) + d dt∣ f(X+ t (Y − X)) (517)t=0→Y −Xwhere df(X) is the directional derivative 3.10 [161] [252] 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) −→Y −Xdf(X) ≽0 ⇔〈〉→Y −Xf(Y ) − f(X) − df(X) , w ≥ 0 ∀w ≽ 0whereR M +→Y −Xdf(X) =⎡⎢⎣R M +(518)tr ( ∇f 1 (X) T (Y − X) ) ⎤tr ( ∇f 2 (X) T (Y − X) )⎥.tr ( ∇f M (X) T (Y − X) ) ⎦ ∈ RM (519)Necessary and sufficient discretization (421) allows relaxation of thesemi-infinite number of conditions w ≽ 0 instead to w ∈ {e i , i=1... M}the extreme directions of the nonnegative orthant. Each extreme direction→Y −Xpicks out a real entry f i and df(X) ifrom vector-valued function f and its→Y −Xdirectional derivative df(X) , then Theorem 3.1.9.0.1 applies.The vector-valued function case (517) is therefore a straightforwardapplication of the first-order convexity condition for real functions to eachentry of the vector-valued function.3.10 We extend the traditional definition of directional derivative inD.1.4 so that directionmay be indicated by a vector or a matrix, thereby broadening the scope of the Taylorseries (D.1.7). The right-hand side of the inequality (517) is the first-order Taylor seriesexpansion of f about X .