Getting the final solution for the subgradient of function $F(x) := max 0, frac12(x^2 - 1)$

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I have to find the subgradients of the following function.

$$F(x) := max left0, frac12(x^2 - 1)right$$

Analytically I can see subdifferentials at $x=-1$ is $nabla f(-1) in [-1 ,0] $ and at $x =1$ is $nabla f(1) in [0,1]$.

I am facing difficulties while obtaining these subdifferentials ($v$) using following inequality, $f(x) - f(bar x) ge langle v,x-bar x rangle, xin R$.

If I apply $bar x = -1$ what should be my $f(x)$?

Similarly what should be the $f(x)$ at $bar x =1$?

How can we obtain above-observed subdifferential using the definition of subdifferential? (above inequality)

convex-analysis convex-optimization subgradient

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edited Jul 30 at 18:45

VHarisop

804421

asked Jul 30 at 10:10

Malintha

1276

• 
  
  
  

  
  

  
  
  
  
  
  

  
  use max, langle and rangle
  – LinAlg
  Jul 30 at 15:25
  

  
  
  
  

add a comment | 

up vote
0
down vote

favorite

I have to find the subgradients of the following function.

$$F(x) := max left0, frac12(x^2 - 1)right$$

Analytically I can see subdifferentials at $x=-1$ is $nabla f(-1) in [-1 ,0] $ and at $x =1$ is $nabla f(1) in [0,1]$.

I am facing difficulties while obtaining these subdifferentials ($v$) using following inequality, $f(x) - f(bar x) ge langle v,x-bar x rangle, xin R$.

If I apply $bar x = -1$ what should be my $f(x)$?

Similarly what should be the $f(x)$ at $bar x =1$?

How can we obtain above-observed subdifferential using the definition of subdifferential? (above inequality)

convex-analysis convex-optimization subgradient

share|cite|improve this question

edited Jul 30 at 18:45

VHarisop

804421

asked Jul 30 at 10:10

Malintha

1276

• 
  
  
  

  
  

  
  
  
  
  
  

  
  use max, langle and rangle
  – LinAlg
  Jul 30 at 15:25
  

  
  
  
  

add a comment | 

up vote
0
down vote

favorite

up vote
0
down vote

favorite

I have to find the subgradients of the following function.

$$F(x) := max left0, frac12(x^2 - 1)right$$

Analytically I can see subdifferentials at $x=-1$ is $nabla f(-1) in [-1 ,0] $ and at $x =1$ is $nabla f(1) in [0,1]$.

I am facing difficulties while obtaining these subdifferentials ($v$) using following inequality, $f(x) - f(bar x) ge langle v,x-bar x rangle, xin R$.

If I apply $bar x = -1$ what should be my $f(x)$?

Similarly what should be the $f(x)$ at $bar x =1$?

How can we obtain above-observed subdifferential using the definition of subdifferential? (above inequality)

convex-analysis convex-optimization subgradient

share|cite|improve this question

edited Jul 30 at 18:45

VHarisop

804421

asked Jul 30 at 10:10

Malintha

1276

I have to find the subgradients of the following function.

$$F(x) := max left0, frac12(x^2 - 1)right$$

Analytically I can see subdifferentials at $x=-1$ is $nabla f(-1) in [-1 ,0] $ and at $x =1$ is $nabla f(1) in [0,1]$.

I am facing difficulties while obtaining these subdifferentials ($v$) using following inequality, $f(x) - f(bar x) ge langle v,x-bar x rangle, xin R$.

If I apply $bar x = -1$ what should be my $f(x)$?

Similarly what should be the $f(x)$ at $bar x =1$?

How can we obtain above-observed subdifferential using the definition of subdifferential? (above inequality)

convex-analysis convex-optimization subgradient

share|cite|improve this question

edited Jul 30 at 18:45

VHarisop

804421

asked Jul 30 at 10:10

Malintha

1276

share|cite|improve this question

share|cite|improve this question

edited Jul 30 at 18:45

VHarisop

804421

edited Jul 30 at 18:45

VHarisop

804421

edited Jul 30 at 18:45

VHarisop

804421

804421

asked Jul 30 at 10:10

Malintha

1276

asked Jul 30 at 10:10

Malintha

1276

asked Jul 30 at 10:10

Malintha

1276

1276

• 
  
  
  

  
  

  
  
  
  
  
  

  
  use max, langle and rangle
  – LinAlg
  Jul 30 at 15:25
  

  
  
  
  

add a comment | 

• 
  
  
  

  
  

  
  
  
  
  
  

  
  use max, langle and rangle
  – LinAlg
  Jul 30 at 15:25
  

  
  
  
  

use max, langle and rangle
– LinAlg
Jul 30 at 15:25

use max, langle and rangle
– LinAlg
Jul 30 at 15:25

add a comment | 

1 Answer
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For $barx=-1$ you get:
$$f(x) - f(-1) geq langle v,x-(-1) rangle$$
$$f(x) geq langle v,x+1 rangle$$
Clearly $v=0$ is a subdifferential, because $f(x) geq 0$ for all $x in mathbbR$. But also $v=-1$ is a subdifferential because $f(x) geq -x-1$ for all $x in mathbbR$ (just draw a plot of $F$ and of $g(x) = -x-1$).

share|cite|improve this answer

answered Jul 30 at 15:31

LinAlg

5,4111319

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1 Answer
1

active

oldest

votes

1 Answer
1

active

oldest

votes

active

oldest

votes

active

oldest

votes

up vote
1
down vote



accepted

For $barx=-1$ you get:
$$f(x) - f(-1) geq langle v,x-(-1) rangle$$
$$f(x) geq langle v,x+1 rangle$$
Clearly $v=0$ is a subdifferential, because $f(x) geq 0$ for all $x in mathbbR$. But also $v=-1$ is a subdifferential because $f(x) geq -x-1$ for all $x in mathbbR$ (just draw a plot of $F$ and of $g(x) = -x-1$).

share|cite|improve this answer

answered Jul 30 at 15:31

LinAlg

5,4111319

add a comment | 

up vote
1
down vote



accepted

For $barx=-1$ you get:
$$f(x) - f(-1) geq langle v,x-(-1) rangle$$
$$f(x) geq langle v,x+1 rangle$$
Clearly $v=0$ is a subdifferential, because $f(x) geq 0$ for all $x in mathbbR$. But also $v=-1$ is a subdifferential because $f(x) geq -x-1$ for all $x in mathbbR$ (just draw a plot of $F$ and of $g(x) = -x-1$).

share|cite|improve this answer

answered Jul 30 at 15:31

LinAlg

5,4111319

add a comment | 

up vote
1
down vote



accepted

up vote
1
down vote



accepted

For $barx=-1$ you get:
$$f(x) - f(-1) geq langle v,x-(-1) rangle$$
$$f(x) geq langle v,x+1 rangle$$
Clearly $v=0$ is a subdifferential, because $f(x) geq 0$ for all $x in mathbbR$. But also $v=-1$ is a subdifferential because $f(x) geq -x-1$ for all $x in mathbbR$ (just draw a plot of $F$ and of $g(x) = -x-1$).

share|cite|improve this answer

answered Jul 30 at 15:31

LinAlg

5,4111319

For $barx=-1$ you get:
$$f(x) - f(-1) geq langle v,x-(-1) rangle$$
$$f(x) geq langle v,x+1 rangle$$
Clearly $v=0$ is a subdifferential, because $f(x) geq 0$ for all $x in mathbbR$. But also $v=-1$ is a subdifferential because $f(x) geq -x-1$ for all $x in mathbbR$ (just draw a plot of $F$ and of $g(x) = -x-1$).

share|cite|improve this answer

answered Jul 30 at 15:31

LinAlg

5,4111319

share|cite|improve this answer

share|cite|improve this answer

answered Jul 30 at 15:31

LinAlg

5,4111319

answered Jul 30 at 15:31

LinAlg

5,4111319

answered Jul 30 at 15:31

LinAlg

5,4111319

5,4111319

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add a comment | 

 

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