March 2014
Intermediate to advanced
412 pages
11h 16m
English
Content preview from Numerical Methods and Optimization
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Basic Concepts 171
x
1
x
2
y
∇f(¯x)
¯x
x
(
¯
t)
y = f(x
1
,x
2
)
y = c
x = x(t)
FIGURE 8.8: A level set and a gradient.
So, if we denote by ¯x = x(
¯
t), then from the last two equations we obtain
∇f(¯x)
T
x
(
¯
t)=0. (8.12)
Geometrically, this means that vectors ∇f (¯x)andx
(
¯
t) are orthogonal. Note
that x
(
¯
t) represents the tangent line to x(t)at¯x. Thus, we have the following
property. Let f (x) be a continuously differentiable function, and let x(t)bea
continuously differentiable curve passing through ¯x in the level set of f (x)at
the level c = f (¯x), where x(
¯
t)=¯x, x
(
¯
t) = 0. Then the gradient of f at ¯x is
orthogonal to the tangent line of x(t)at¯x. This is illustrated ...
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