v2007.09.13 - Convex Optimization

D.1. DIRECTIONAL DERIVATIVE, TAYLOR SERIES 557D.1.4First directional derivativeAssume that a differentiable function g(X) : R K×L →R M×N has continuousfirst- and second-order gradients ∇g and ∇ 2 g over domg which is an openset. We seek simple expressions for the first and second directional derivativesin direction Y ∈R K×L →Y, dg ∈ R M×N and dg →Y2 ∈ R M×N respectively.Assuming that the limit exists, we may state the partial derivative of themn th entry of g with respect to the kl th entry of X ;∂g mn (X)∂X klg mn (X + ∆t e= limk e T l ) − g mn(X)∆t→0 ∆t∈ R (1566)where e k is the k th standard basis vector in R K while e l is the l th standardbasis vector in R L . The total number of partial derivatives equals KLMNwhile the gradient is defined in their terms; the mn th entry of the gradient is⎡∇g mn (X) =⎢⎣∂g mn(X)∂X 11∂g mn(X)∂X 21.∂g mn(X)∂X K1∂g mn(X)∂X 12· · ·∂g mn(X)∂X 22· · ·.∂g mn(X)∂X K2· · ·∂g mn(X)∂X 1L∂g mn(X)∂X 2L.∂g mn(X)∂X KL⎤∈ R K×L (1567)⎥⎦while the gradient is a quartix⎡∇g(X) = ⎢⎣∇g 11 (X) ∇g 12 (X) · · · ∇g 1N (X)∇g 21 (X) ∇g 22 (X) · · · ∇g 2N (X). .∇g M1 (X) ∇g M2 (X) · · ·.∇g MN (X)⎤⎥⎦ ∈ RM×N×K×L (1568)By simply rotating our perspective of the four-dimensional representation ofgradient matrix, we find one of three useful transpositions of this quartix(connoted T 1 ):