v2007.09.13 - Convex Optimization

552 APPENDIX D. MATRIX CALCULUSGradient of vector-valued function g(X) : R K×L →R N on matrix domainis a cubix[∇X(:,1) g 1 (X) ∇ X(:,1) g 2 (X) · · · ∇ X(:,1) g N (X)∇g(X) ∆ =∇ X(:,2) g 1 (X) ∇ X(:,2) g 2 (X) · · · ∇ X(:,2) g N (X).........∇ X(:,L) g 1 (X) ∇ X(:,L) g 2 (X) · · · ∇ X(:,L) g N (X) ]= [ ∇g 1 (X) ∇g 2 (X) · · · ∇g N (X) ] ∈ R K×N×L (1539)while the second-order gradient has a five-dimensional representation;[∇∇X(:,1) g 1 (X) ∇∇ X(:,1) g 2 (X) · · · ∇∇ X(:,1) g N (X)∇ 2 g(X) ∆ =∇∇ X(:,2) g 1 (X) ∇∇ X(:,2) g 2 (X) · · · ∇∇ X(:,2) g N (X).........∇∇ X(:,L) g 1 (X) ∇∇ X(:,L) g 2 (X) · · · ∇∇ X(:,L) g N (X) ]= [ ∇ 2 g 1 (X) ∇ 2 g 2 (X) · · · ∇ 2 g N (X) ] ∈ R K×N×L×K×L (1540)The gradient of matrix-valued function g(X) : R K×L →R M×N on matrixdomain has a four-dimensional representation called 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 (1541)while the second-order gradient has six-dimensional representation⎡∇ 2 g(X) =∆ ⎢⎣and so on.∇ 2 g 11 (X) ∇ 2 g 12 (X) · · · ∇ 2 g 1N (X)∇ 2 g 21 (X) ∇ 2 g 22 (X) · · · ∇ 2 g 2N (X). .∇ 2 g M1 (X) ∇ 2 g M2 (X) · · ·.∇ 2 g MN (X)⎤⎥⎦ ∈ RM×N×K×L×K×L(1542)