v2010.10.26 - Convex Optimization

3.6. GRADIENT 2613.6.4 second-order convexity conditionAgain, by discretization (498), we are obliged only to consider each individualentry f i of a vector-valued function f ; id est, the real functions {f i }.For f(X) : R p →R M , a twice differentiable vector-valued function withvector argument on open convex domain,∇ 2 f i (X) ≽0 ∀X ∈ domf , i=1... M (617)S p +is a necessary and sufficient condition for convexity of f . Obviously,when M = 1, this convexity condition also serves for a real function.Condition (617) demands nonnegative curvature, intuitively, henceprecluding points of inflection as in Figure 79 (p.270).Strict inequality in (617) provides only a sufficient condition for strictconvexity, but that is nothing new; videlicet, strictly convex real functionf i (x)=x 4 does not have positive second derivative at each and every x∈ R .Quadratic forms constitute a notable exception where the strict-case converseholds reliably.3.6.4.0.1 Example. Convex quadratic.Real quadratic multivariate polynomial in matrix A and vector bx T Ax + 2b T x + c (618)is convex if and only if A≽0.gradient: (D.2.1)Proof follows by observing second order∇ 2 x(x T Ax + 2b T x + c ) = A +A T (619)Because x T (A +A T )x = 2x T Ax, matrix A can be assumed symmetric. 3.6.4.0.2 Exercise. Real fractional function. (confer3.3,3.5.0.0.4)Prove that real function f(x,y) = x/y is not convex on the first quadrant.Also exhibit this in a plot of the function. (f is quasilinear (p.271) on{y > 0} , in fact, and nonmonotonic; even on the first quadrant.)