v2007.09.17 - Convex Optimization

208 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONSf(Y )[ ∇f(X)−1]∂H −Figure 59: When a real function f is differentiable at each point in its opendomain, there is an intuitive geometric interpretation of function convexityin terms of its gradient ∇f and its epigraph: Drawn is a convex quadraticbowl in R 2 ×R (confer Figure 117, p.563); f(Y )= Y T Y : R 2 → R versus Yon some open disc in R 2 . Supporting hyperplane ∂H − ∈ R 2 × R (which istangent, only partially drawn) and its normal vector [ ∇f(X) T −1 ] T at theparticular 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.