Convex optimization:

please help me understand this subject with full answer. thanks alot (:

3.9 Second-order conditions for convexity on an affine set. Let FERnam, x E R". The restriction of f : R" - R to the affine set { Fz + x | z E R" } is defined as the function f : Rm - R with f ( z ) = f(Fz + 3), dom f = {z | Fz + x ( dom f}. Suppose f is twice differentiable with a convex domain. (a) Show that f is convex if and only if for all z E dom f FTV2 f ( F z + 3 ) F Z O. (b) Suppose A E RPX" is a matrix whose nullspace is equal to the range of F, i.e., AF = 0 and rank A = n - rank F. Show that f is convex if and only if for all z ( dom f there exists a ) E R such that V2 f ( F z + i ) + XAT AZ O. Hint. Use the following result: If B E S" and A E RPX", then xBx > 0 for all x EN(A) if and only if there exists a A such that B + AAT A > 0