positive linear combination of quasi-concave functions
$begingroup$
I have a question that I cannot manage to get around. I need to answer the following:
Give an example to 2 quasi-concave functions on an interval such that
any positive linear combination of these two functions is not
quasi-concave.
Now, I understand the property that states that concavity is preserved by negative linear combinations. So my first question is, is the contrary always true? That is, that concavity is not respected by positive linear combinations?
If this is not the case always, my main issue here is how to get an example that clearly shows the above for ANY positive linear combination. I imagine this would require an example and a proof but I cannot work around this and to the best of my knowledge this is not covered by another thread.
Thanks in advance, any help would be useful
convex-analysis
asked Dec 9 '18 at 15:29
user20105user20105
82
$endgroup$
add a comment |
$begingroup$
I have a question that I cannot manage to get around. I need to answer the following:
Give an example to 2 quasi-concave functions on an interval such that
any positive linear combination of these two functions is not
quasi-concave.
Now, I understand the property that states that concavity is preserved by negative linear combinations. So my first question is, is the contrary always true? That is, that concavity is not respected by positive linear combinations?
If this is not the case always, my main issue here is how to get an example that clearly shows the above for ANY positive linear combination. I imagine this would require an example and a proof but I cannot work around this and to the best of my knowledge this is not covered by another thread.
Thanks in advance, any help would be useful
convex-analysis
asked Dec 9 '18 at 15:29
user20105user20105
82
$endgroup$
add a comment |
$begingroup$
I have a question that I cannot manage to get around. I need to answer the following:
Give an example to 2 quasi-concave functions on an interval such that
any positive linear combination of these two functions is not
quasi-concave.
Now, I understand the property that states that concavity is preserved by negative linear combinations. So my first question is, is the contrary always true? That is, that concavity is not respected by positive linear combinations?
If this is not the case always, my main issue here is how to get an example that clearly shows the above for ANY positive linear combination. I imagine this would require an example and a proof but I cannot work around this and to the best of my knowledge this is not covered by another thread.
Thanks in advance, any help would be useful
convex-analysis
asked Dec 9 '18 at 15:29
user20105user20105
82
$endgroup$
I have a question that I cannot manage to get around. I need to answer the following:
Give an example to 2 quasi-concave functions on an interval such that
any positive linear combination of these two functions is not
quasi-concave.
Now, I understand the property that states that concavity is preserved by negative linear combinations. So my first question is, is the contrary always true? That is, that concavity is not respected by positive linear combinations?
If this is not the case always, my main issue here is how to get an example that clearly shows the above for ANY positive linear combination. I imagine this would require an example and a proof but I cannot work around this and to the best of my knowledge this is not covered by another thread.
Thanks in advance, any help would be useful
convex-analysis
convex-analysis
asked Dec 9 '18 at 15:29
user20105user20105
82
asked Dec 9 '18 at 15:29
user20105user20105
82
asked Dec 9 '18 at 15:29
user20105user20105
82
asked Dec 9 '18 at 15:29
user20105user20105
82
asked Dec 9 '18 at 15:29
user20105user20105
82
82
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
The function $xmapsto sqrt |x|$ is quasi-convex. Let me show that the function
$$
f(x) = a sqrt{|x-1|} + b sqrt{|x+1|}
$$
is not quasi-convex for all $a,b>0$.
The points $x=-1$ and $x=1$ are local minima of $f$.
On the interval $(-1,1)$ the function $f$ reduces to
$$
f(x) = a sqrt{1-x} + b sqrt{x+1},
$$
which is a strictly concave function. Hence, $f$ has a local maximum $x^*in (-1,1)$ with $f(x^*) > max (f(-1),f(1))$.
Now let me choose a sub-level set that contains $1$ and $-1$ but not $x^*$.
Then the sub-level set
$$
left{x : f(x) le frac{f(x^*)+max (f(-1),f(1))}2right}
$$
contains $-1,1$ but not $x^*$. Hence this level set is not convex, and $f$ is not quasi-convex.
Note that $-f$ is not quasi-concave, but is the sum of two quasi-concave functions. So is is the example you are looking for.
answered Dec 11 '18 at 8:03
dawdaw
24.2k1545
$endgroup$
$begingroup$
Thanks for your answer but I am failing to see how this answers the question, would you mind further explaining?
$endgroup$
– user20105
Dec 13 '18 at 14:56
$begingroup$
I expanded the answer. Could you be more specific? What argument is difficult to understand?
$endgroup$
– daw
Dec 13 '18 at 15:06
$begingroup$
I am looking for two quasi-concave functions on an interval such that any positive linear combination of these is not quasi-concave. Here on the interval (-1,1) the function f(x) is the sum of two concave functions, it is concave and the level set is not convex. I apologize for the confusion but how is f(x) an example of a linear combination of two quasiconcave functions with f(x) not quasi-concave?
$endgroup$
– user20105
Dec 13 '18 at 15:31
$begingroup$
The function $-f$ on the interval $[-2,2]$ is the counterexample: it is the sum of two quasi-concave functions, but not quasi-concave. Note the minus sign. It was easier for me two write the example for quasi-convex. To get to quasi-concave, you have multiply by $-1$ everything.
$endgroup$
– daw
Dec 13 '18 at 15:41
$begingroup$
I see it! Thank you for the answer and the explanation!
$endgroup$
– user20105
Dec 13 '18 at 15:47
add a comment |
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1 Answer
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1 Answer
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$begingroup$
The function $xmapsto sqrt |x|$ is quasi-convex. Let me show that the function
$$
f(x) = a sqrt{|x-1|} + b sqrt{|x+1|}
$$
is not quasi-convex for all $a,b>0$.
The points $x=-1$ and $x=1$ are local minima of $f$.
On the interval $(-1,1)$ the function $f$ reduces to
$$
f(x) = a sqrt{1-x} + b sqrt{x+1},
$$
which is a strictly concave function. Hence, $f$ has a local maximum $x^*in (-1,1)$ with $f(x^*) > max (f(-1),f(1))$.
Now let me choose a sub-level set that contains $1$ and $-1$ but not $x^*$.
Then the sub-level set
$$
left{x : f(x) le frac{f(x^*)+max (f(-1),f(1))}2right}
$$
contains $-1,1$ but not $x^*$. Hence this level set is not convex, and $f$ is not quasi-convex.
Note that $-f$ is not quasi-concave, but is the sum of two quasi-concave functions. So is is the example you are looking for.
answered Dec 11 '18 at 8:03
dawdaw
24.2k1545
$endgroup$
$begingroup$
Thanks for your answer but I am failing to see how this answers the question, would you mind further explaining?
$endgroup$
– user20105
Dec 13 '18 at 14:56
$begingroup$
I expanded the answer. Could you be more specific? What argument is difficult to understand?
$endgroup$
– daw
Dec 13 '18 at 15:06
$begingroup$
I am looking for two quasi-concave functions on an interval such that any positive linear combination of these is not quasi-concave. Here on the interval (-1,1) the function f(x) is the sum of two concave functions, it is concave and the level set is not convex. I apologize for the confusion but how is f(x) an example of a linear combination of two quasiconcave functions with f(x) not quasi-concave?
$endgroup$
– user20105
Dec 13 '18 at 15:31
$begingroup$
The function $-f$ on the interval $[-2,2]$ is the counterexample: it is the sum of two quasi-concave functions, but not quasi-concave. Note the minus sign. It was easier for me two write the example for quasi-convex. To get to quasi-concave, you have multiply by $-1$ everything.
$endgroup$
– daw
Dec 13 '18 at 15:41
$begingroup$
I see it! Thank you for the answer and the explanation!
$endgroup$
– user20105
Dec 13 '18 at 15:47
add a comment |
$begingroup$
The function $xmapsto sqrt |x|$ is quasi-convex. Let me show that the function
$$
f(x) = a sqrt{|x-1|} + b sqrt{|x+1|}
$$
is not quasi-convex for all $a,b>0$.
The points $x=-1$ and $x=1$ are local minima of $f$.
On the interval $(-1,1)$ the function $f$ reduces to
$$
f(x) = a sqrt{1-x} + b sqrt{x+1},
$$
which is a strictly concave function. Hence, $f$ has a local maximum $x^*in (-1,1)$ with $f(x^*) > max (f(-1),f(1))$.
Now let me choose a sub-level set that contains $1$ and $-1$ but not $x^*$.
Then the sub-level set
$$
left{x : f(x) le frac{f(x^*)+max (f(-1),f(1))}2right}
$$
contains $-1,1$ but not $x^*$. Hence this level set is not convex, and $f$ is not quasi-convex.
Note that $-f$ is not quasi-concave, but is the sum of two quasi-concave functions. So is is the example you are looking for.
answered Dec 11 '18 at 8:03
dawdaw
24.2k1545
$endgroup$
$begingroup$
Thanks for your answer but I am failing to see how this answers the question, would you mind further explaining?
$endgroup$
– user20105
Dec 13 '18 at 14:56
$begingroup$
I expanded the answer. Could you be more specific? What argument is difficult to understand?
$endgroup$
– daw
Dec 13 '18 at 15:06
$begingroup$
I am looking for two quasi-concave functions on an interval such that any positive linear combination of these is not quasi-concave. Here on the interval (-1,1) the function f(x) is the sum of two concave functions, it is concave and the level set is not convex. I apologize for the confusion but how is f(x) an example of a linear combination of two quasiconcave functions with f(x) not quasi-concave?
$endgroup$
– user20105
Dec 13 '18 at 15:31
$begingroup$
The function $-f$ on the interval $[-2,2]$ is the counterexample: it is the sum of two quasi-concave functions, but not quasi-concave. Note the minus sign. It was easier for me two write the example for quasi-convex. To get to quasi-concave, you have multiply by $-1$ everything.
$endgroup$
– daw
Dec 13 '18 at 15:41
$begingroup$
I see it! Thank you for the answer and the explanation!
$endgroup$
– user20105
Dec 13 '18 at 15:47
add a comment |
$begingroup$
The function $xmapsto sqrt |x|$ is quasi-convex. Let me show that the function
$$
f(x) = a sqrt{|x-1|} + b sqrt{|x+1|}
$$
is not quasi-convex for all $a,b>0$.
The points $x=-1$ and $x=1$ are local minima of $f$.
On the interval $(-1,1)$ the function $f$ reduces to
$$
f(x) = a sqrt{1-x} + b sqrt{x+1},
$$
which is a strictly concave function. Hence, $f$ has a local maximum $x^*in (-1,1)$ with $f(x^*) > max (f(-1),f(1))$.
Now let me choose a sub-level set that contains $1$ and $-1$ but not $x^*$.
Then the sub-level set
$$
left{x : f(x) le frac{f(x^*)+max (f(-1),f(1))}2right}
$$
contains $-1,1$ but not $x^*$. Hence this level set is not convex, and $f$ is not quasi-convex.
Note that $-f$ is not quasi-concave, but is the sum of two quasi-concave functions. So is is the example you are looking for.
answered Dec 11 '18 at 8:03
dawdaw
24.2k1545
$endgroup$
The function $xmapsto sqrt |x|$ is quasi-convex. Let me show that the function
$$
f(x) = a sqrt{|x-1|} + b sqrt{|x+1|}
$$
is not quasi-convex for all $a,b>0$.
The points $x=-1$ and $x=1$ are local minima of $f$.
On the interval $(-1,1)$ the function $f$ reduces to
$$
f(x) = a sqrt{1-x} + b sqrt{x+1},
$$
which is a strictly concave function. Hence, $f$ has a local maximum $x^*in (-1,1)$ with $f(x^*) > max (f(-1),f(1))$.
Now let me choose a sub-level set that contains $1$ and $-1$ but not $x^*$.
Then the sub-level set
$$
left{x : f(x) le frac{f(x^*)+max (f(-1),f(1))}2right}
$$
contains $-1,1$ but not $x^*$. Hence this level set is not convex, and $f$ is not quasi-convex.
Note that $-f$ is not quasi-concave, but is the sum of two quasi-concave functions. So is is the example you are looking for.
answered Dec 11 '18 at 8:03
dawdaw
24.2k1545
edited Dec 13 '18 at 15:06
answered Dec 11 '18 at 8:03
dawdaw
24.2k1545
answered Dec 11 '18 at 8:03
dawdaw
24.2k1545
answered Dec 11 '18 at 8:03
dawdaw
24.2k1545
24.2k1545
$begingroup$
Thanks for your answer but I am failing to see how this answers the question, would you mind further explaining?
$endgroup$
– user20105
Dec 13 '18 at 14:56
$begingroup$
I expanded the answer. Could you be more specific? What argument is difficult to understand?
$endgroup$
– daw
Dec 13 '18 at 15:06
$begingroup$
I am looking for two quasi-concave functions on an interval such that any positive linear combination of these is not quasi-concave. Here on the interval (-1,1) the function f(x) is the sum of two concave functions, it is concave and the level set is not convex. I apologize for the confusion but how is f(x) an example of a linear combination of two quasiconcave functions with f(x) not quasi-concave?
$endgroup$
– user20105
Dec 13 '18 at 15:31
$begingroup$
The function $-f$ on the interval $[-2,2]$ is the counterexample: it is the sum of two quasi-concave functions, but not quasi-concave. Note the minus sign. It was easier for me two write the example for quasi-convex. To get to quasi-concave, you have multiply by $-1$ everything.
$endgroup$
– daw
Dec 13 '18 at 15:41
$begingroup$
I see it! Thank you for the answer and the explanation!
$endgroup$
– user20105
Dec 13 '18 at 15:47
add a comment |
$begingroup$
Thanks for your answer but I am failing to see how this answers the question, would you mind further explaining?
$endgroup$
– user20105
Dec 13 '18 at 14:56
$begingroup$
I expanded the answer. Could you be more specific? What argument is difficult to understand?
$endgroup$
– daw
Dec 13 '18 at 15:06
$begingroup$
I am looking for two quasi-concave functions on an interval such that any positive linear combination of these is not quasi-concave. Here on the interval (-1,1) the function f(x) is the sum of two concave functions, it is concave and the level set is not convex. I apologize for the confusion but how is f(x) an example of a linear combination of two quasiconcave functions with f(x) not quasi-concave?
$endgroup$
– user20105
Dec 13 '18 at 15:31
$begingroup$
The function $-f$ on the interval $[-2,2]$ is the counterexample: it is the sum of two quasi-concave functions, but not quasi-concave. Note the minus sign. It was easier for me two write the example for quasi-convex. To get to quasi-concave, you have multiply by $-1$ everything.
$endgroup$
– daw
Dec 13 '18 at 15:41
$begingroup$
I see it! Thank you for the answer and the explanation!
$endgroup$
– user20105
Dec 13 '18 at 15:47
$begingroup$
Thanks for your answer but I am failing to see how this answers the question, would you mind further explaining?
$endgroup$
– user20105
Dec 13 '18 at 14:56
$begingroup$
Thanks for your answer but I am failing to see how this answers the question, would you mind further explaining?
$endgroup$
– user20105
Dec 13 '18 at 14:56
$begingroup$
I expanded the answer. Could you be more specific? What argument is difficult to understand?
$endgroup$
– daw
Dec 13 '18 at 15:06
$begingroup$
I expanded the answer. Could you be more specific? What argument is difficult to understand?
$endgroup$
– daw
Dec 13 '18 at 15:06
$begingroup$
I am looking for two quasi-concave functions on an interval such that any positive linear combination of these is not quasi-concave. Here on the interval (-1,1) the function f(x) is the sum of two concave functions, it is concave and the level set is not convex. I apologize for the confusion but how is f(x) an example of a linear combination of two quasiconcave functions with f(x) not quasi-concave?
$endgroup$
– user20105
Dec 13 '18 at 15:31
$begingroup$
I am looking for two quasi-concave functions on an interval such that any positive linear combination of these is not quasi-concave. Here on the interval (-1,1) the function f(x) is the sum of two concave functions, it is concave and the level set is not convex. I apologize for the confusion but how is f(x) an example of a linear combination of two quasiconcave functions with f(x) not quasi-concave?
$endgroup$
– user20105
Dec 13 '18 at 15:31
$begingroup$
The function $-f$ on the interval $[-2,2]$ is the counterexample: it is the sum of two quasi-concave functions, but not quasi-concave. Note the minus sign. It was easier for me two write the example for quasi-convex. To get to quasi-concave, you have multiply by $-1$ everything.
$endgroup$
– daw
Dec 13 '18 at 15:41
$begingroup$
The function $-f$ on the interval $[-2,2]$ is the counterexample: it is the sum of two quasi-concave functions, but not quasi-concave. Note the minus sign. It was easier for me two write the example for quasi-convex. To get to quasi-concave, you have multiply by $-1$ everything.
$endgroup$
– daw
Dec 13 '18 at 15:41
$begingroup$
I see it! Thank you for the answer and the explanation!
$endgroup$
– user20105
Dec 13 '18 at 15:47
$begingroup$
I see it! Thank you for the answer and the explanation!
$endgroup$
– user20105
Dec 13 '18 at 15:47
add a comment |
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