v2010.10.26 - Convex Optimization

262 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS3.6.4.0.3 Exercise. Stress function.Define |x −y| √ (x −y) 2 andX = [x 1 · · · x N ] ∈ R 1×N (76)Given symmetric nonnegative data [h ij ] ∈ S N ∩ R N×N+ , consider functionN−1∑ N∑f(vec X) = (|x i − x j | − h ij ) 2 ∈ R (1308)i=1 j=i+1Take the gradient and Hessian of f . Then explain why f is not a convexfunction; id est, why doesn’t second-order condition (617) apply to theconstant positive semidefinite Hessian matrix you found. For N = 6 and h ijdata from (1381), apply line theorem 3.7.3.0.1 to plot f along some arbitrarylines through its domain.3.6.4.1 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 ∈ domf i (D.1.7) to threeterms: On some open interval of ‖Y ‖ 2 , so that each and every line segment[X,Y ] belongs to domf i , there exists an α∈[0, 1] such that [391,1.2.3][42,1.1.4]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)(620)The first-order condition for convexity (608) follows directly from this andthe second-order condition (617).3.7 Convex matrix-valued 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 (621) with respectto the positive semidefinite cone S M + in the ambient space of symmetricmatrices. 3.223.22 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