v2007.09.13 - Convex Optimization

214 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONSis a necessary and sufficient condition for convexity of f . Obviously, whenM = 1, this convexity condition also serves for a real function. Intuitively,condition (519) precludes points of inflection, as in Figure 61 on page 220.Strict inequality is a sufficient condition for strict convexity, but that isnothing new; videlicet, the strictly convex real function f i (x)=x 4 does nothave positive second derivative at each and every x∈ R . Quadratic formsconstitute a notable exception where the strict-case converse is reliably true.3.1.11.0.1 Exercise. Real fractional function. (confer3.1.4,3.1.7.0.4)Prove that real function f(x,y) = x/y is not convex on the nonnegativeorthant. Also exhibit this in a plot of the function. (In fact, f is quasilinear(p.222) on {y > 0}.)3.1.11.1 second-order ⇒ first-order conditionFor a twice-differentiable real function f i (X) : R p →R having open domain,a consequence of the mean value theorem from calculus allows compressionof its complete Taylor series expansion about X ∈ dom f i (D.1.7) to threeterms: On some open interval of ‖Y ‖ so each and every line segment[X,Y ] belongs to domf i , there exists an α∈[0, 1] such that [296,1.2.3][30,1.1.4]f i (Y ) = f i (X)+ ∇f i (X) T (Y −X)+ 1 2 (Y −X)T ∇ 2 f i (αX +(1 −α)Y )(Y −X)(520)The first-order condition for convexity (510) follows directly from this andthe second-order condition (519).3.2 Matrix-valued convex functionWe need different tools for matrix argument: We are primarily interested incontinuous matrix-valued functions g(X). We choose symmetric g(X)∈ S Mbecause matrix-valued functions are most often compared (521) with respectto the positive semidefinite cone S M + in the ambient space of symmetricmatrices. 3.103.10 Function symmetry is not a necessary requirement for convexity; indeed, for A∈R m×pand B ∈R m×k , g(X) = AX + B is a convex (affine) function in X on domain R p×k with