v2009.01.01 - Convex Optimization

3.1. CONVEX FUNCTION 221
2
1.5
1
0.5
Y 2
0
−0.5
−1
−1.5
−2
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2
Figure 66: Gradient in R 2 evaluated on grid over some open disc in domain
of convex quadratic bowl f(Y )= Y T Y : R 2 → R illustrated in Figure 67.
Circular contours are level sets; each defined by a constant function-value.
Y 1
3.1.8 gradient
Gradient ∇f of any differentiable multidimensional function f (formally
defined inD.1) maps each entry f i to a space having the same dimension
as the ambient space of its domain. Notation ∇f is shorthand for gradient
∇ x f(x) of f with respect to x . ∇f(y) can mean ∇ y f(y) or gradient
∇ x f(y) of f(x) with respect to x evaluated at y ; a distinction that should
become clear from context.
Gradient of a differentiable real function f(x) : R K →R with respect to
its vector argument is defined
⎡
∇f(x) =
∆ ⎢
⎣
∂f(x)
∂x 1
∂f(x)
∂x 2
.
∂f(x)
∂x K
⎤
⎥
⎦ ∈ RK (1637)
while the second-order gradient of the twice differentiable real function with