v2004.06.19 - Convex Optimization

D.2. TABLES OF GRADIENTS AND DERIVATIVES 373D.2.1Algebraic∇ x x = ∇ x x T = I ∈ R k×k ∇ X X = ∇ X X T = ∆ I ∈ R K×L×K×L (the identity)∇ x (Ax − b) = A T(∇ x x T A − b T) = A∇ x (Ax − b) T (Ax − b) = 2A T (Ax − b)∇x(Ax 2 − b) T (Ax − b) = 2A T A(∇ x x T Ax + 2x T By + y T Cy ) = (A +A T )x + 2By∇ 2 x(x T Ax + 2x T By + y T Cy ) = A +A T ∇ X a T Xb = ∇ X b T X T a = ab T∇ X a T X 2 b = X T ab T + ab T X T∇ X a T X −1 b = −X −T ab T X −T∇ x a T x T xb = 2xa T b ∇ X a T X T Xb = X(ab T + ba T )∇ x a T xx T b = (ab T + ba T )x ∇ X a T XX T b = (ab T + ba T )X∇ X (X −1 ) kl = ∂X−1∂X kl= −X −1 E kl X −1 , confer (1105)(1164)∇ x a T x T xa = 2xa T a∇ x a T xx T a = 2aa T x∇ x a T yx T b = ba T y∇ x a T y T xb = yb T a∇ x a T xy T b = ab T y∇ x a T x T yb = ya T b∇ X a T X T Xa = 2Xaa T∇ X a T XX T a = 2aa T X∇ X a T YX T b = ba T Y∇ X a T Y T Xb = Y ab T∇ X a T XY T b = ab T Y∇ X a T X T Y b = Y ba T