Chapter 3 Geometry of convex functions - Meboo Publishing ...
Chapter 3 Geometry of convex functions - Meboo Publishing ...
Chapter 3 Geometry of convex functions - Meboo Publishing ...
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3.7. GRADIENT 241<br />
2<br />
1.5<br />
1<br />
0.5<br />
Y 2<br />
0<br />
−0.5<br />
−1<br />
−1.5<br />
−2<br />
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2<br />
Figure 74: Gradient in R 2 evaluated on grid over some open disc in domain<br />
<strong>of</strong> <strong>convex</strong> quadratic bowl f(Y )= Y T Y : R 2 → R illustrated in Figure 75.<br />
Circular contours are level sets; each defined by a constant function-value.<br />
Y 1<br />
3.7 Gradient<br />
Gradient ∇f <strong>of</strong> any differentiable multidimensional function f (formally<br />
defined inD.1) maps each entry f i to a space having the same dimension<br />
as the ambient space <strong>of</strong> its domain. Notation ∇f is shorthand for gradient<br />
∇ x f(x) <strong>of</strong> f with respect to x . ∇f(y) can mean ∇ y f(y) or gradient<br />
∇ x f(y) <strong>of</strong> f(x) with respect to x evaluated at y ; a distinction that should<br />
become clear from context.<br />
Gradient <strong>of</strong> a differentiable real function f(x) : R K →R with respect to<br />
its vector argument is defined<br />
⎡<br />
∇f(x) =<br />
⎢<br />
⎣<br />
∂f(x)<br />
∂x 1<br />
∂f(x)<br />
∂x 2<br />
.<br />
∂f(x)<br />
∂x K<br />
⎤<br />
⎥<br />
⎦ ∈ RK (1719)<br />
while the second-order gradient <strong>of</strong> the twice differentiable real function with