v2009.01.01 - Convex Optimization

226 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS
f(Y )
[ ∇f(X)
−1
]
∂H −
Figure 67: When a real function f is differentiable at each point in its
open domain, there is an intuitive geometric interpretation of function
convexity in terms of its gradient ∇f and its epigraph: Drawn is a convex
quadratic bowl in R 2 ×R (confer Figure 136, p.620); f(Y )= Y T Y : R 2 → R
versus Y on some open disc in R 2 . Unique strictly supporting hyperplane
∂H − ∈ R 2 × R (only partially drawn) and its normal vector [ ∇f(X) T −1 ] T
at the particular point of support [X T f(X) ] T are illustrated. The
interpretation: At each and every coordinate Y , there is such a hyperplane
containing [Y T f(Y ) ] T and supporting the epigraph.