v2007.11.26 - Convex Optimization

3.1. CONVEX FUNCTION 211g is convex nonnegatively monotonic and h is convexg is convex nonpositively monotonic and h is concaveand composite function f is concave wheng is concave nonnegatively monotonic and h is concaveg is concave nonpositively monotonic and h is convexwhere ∞ (−∞) is assigned to convex (concave) g when evaluated outsideits domain. When functions are differentiable, these rules are consequent to(1590). Convexity (concavity) of any g is preserved when h is affine. 3.1.9 first-order convexity condition, real functionDiscretization of w ≽0 in (421) invites refocus to the real-valued function:3.1.9.0.1 Theorem. Necessary and sufficient convexity condition.[46,3.1.3] [88,I.5.2] [301,1.2.3] [30,1.2] [249,4.2] [231,3] For realdifferentiable function f(X) : R p×k →R with matrix argument on openconvex domain, the condition (conferD.1.7)f(Y ) ≥ f(X) + 〈∇f(X) , Y − X〉 for each and every X,Y ∈ domf (510)is necessary and sufficient for convexity of f .⋄When f(X) : R p →R is a real differentiable convex function with vectorargument on open convex domain, there is simplification of the first-ordercondition (510); for each and every X,Y ∈ domff(Y ) ≥ f(X) + ∇f(X) T (Y − X) (511)