Lipschitz Constant

Lipschitz Constant
May 18, 2013
1 What is Lipschitz Constant
First consider a single-variable function f(x) for x inside its domain D.
∣ The magnitude of the slope of f(x) for 2 points (x 1 , f(x 1 )) and (x 2 , f(x 2 )) is thus
f(x 1 ) − f(x 2 ) ∣∣∣
∣ x 1 − x 2
A non-negative real number L , which is the smallest upper bound of the slope , ( i.e. the “limiter”
of the slope ) , is called the Lipschitz Constant
L = sup |f(x 1) − f(x 2 )|
= sup
df
|x 1 − x 2 | ∣dx∣
That is, the Lipschitz Constant limit how fast the function can change, in other words, if there is
no Lipschitz Constant, the function can change extremely fast without border, that is, the function
becomes discontinuous
For 2-variable function
For n−variable function
L y = sup |f(x, y 1) − f(x, y 2 )|
|y 1 − y 2 |
∣ L xn = sup
∂f ∣∣∣
∣∂x n
= sup
∂f
∣ ∂y ∣
2 Use of Lipschitz Constant in Numerical Analysis
1

3 Computation of Lipschitz Constant
The Lipschitz Constant can be found using definition or triangle-inequality
EXAMPLE
f = |y|
EXAMPLE
f = √ |y|
∣ L = sup
∂f
∣∣∣ ∣ ∂y ∣ = sup ∂y ∣ ∂y ∣ = sup 1 = 1
∣ L = sup
∂f
∣∣∣∣ ∣ ∂y ∣ = sup ∂ √ |y|
∂y ∣ = sup √ 1 = ∞ /∈ R +
|y|
Thus there is no Lipschitz Constant
EXAMPLE
f = x 2 y , x ∈ [−5, 2] , y ∈ [3, ∞]
L = sup
∂f
∣ ∂y ∣ = sup ∣ ∣ x
2 = 25
EXAMPLE
f = √ x 2 + y 2 √ √
|f(x, y 1 ) − f(x, y 2 )| = | x 2 + y1 2 − x 2 + y2|
2
√ √
√ =
∣ x 2 + y1 2 − x 2 + y2
2 ∣ ·
x2 + y1 2 + √ x 2 + y2
√ 2 x2 + y2 2 + √ x 2 + y2
2
=
≤
=
|y 2 1 − y 2 2|
√
x2 + y 2 2 + √ x 2 + y 2 2
|y 1 + y 2 |
√
x2 + y2 2 + √ |y 1 − y 2 |
x 2 + y2
2
|y 1 | + |y 2 |
√
x2 + y2 2 + √ |y 1 − y 2 |
x 2 + y2
2 } {{ }
≤1
Thus L = 1
≤ 1 · |y 1 − y 2 |
−END−
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