Hi,

i need to understand the full way of solving those problems in convex optimization. thanks alot

3.27 Diagonal elements of Cholesky factor. Each X E S* * has a unique Cholesky factorization X = LL', where L is lower triangular, with Lui > 0. Show that Lui is a concave function of X (with domain S+ +). Hint. Lii can be expressed as Lii = (w - zTY-1z)1/2, where Y 2 W is the leading i x i submatrix of X. Solution. The function f(z, Y) = zY-z with dom f = {(z, Y) | Y > 0} is convex jointly in z and Y. To see this note that (z, Y, t) Eepif Y > 0, so epi f is a convex set. Therefore, w - 2 Yz is a concave function of X. Since the squareroot is an increasing concave function, it follows from the composition rules that Ikk = (w - zTY-1z)1/2 is a concave function of X