How to show negative entropy function $f(x)=xlog(x)$ is strongly convex?
$begingroup$
Let $f: mathbb{R}_+ rightarrow mathbb{R}$ where $f(x)=xlog(x)$. How to show it is strongly convex, i.e.,
Definition: Let $f: mathbb{R}^n rightarrow mathbb{R}$ be differentiable. Then $f$ is strongly convex if $exists$ a positive constant $alpha > 0$ such that
$$
langle nabla f(y) - nabla f(x),y-x rangle geq alpha||y-x||^2 ,,,,,,,,,forall x,y in mathbb{R}^n tag{1}
$$
Following $(1)$ we have $(y-x)log(frac{y}{x})geq alpha(y-x)^2
,,,,,
forall x,y in mathbb{R}_+$.
What $alpha$ satisfies the above and how we can handle the inequality $forall x,y in mathbb{R}_+$ ?
inequality logarithms convex-analysis
asked Jan 17 at 17:52
SaeedSaeed
1,005310
$endgroup$
|
show 1 more comment
$begingroup$
Let $f: mathbb{R}_+ rightarrow mathbb{R}$ where $f(x)=xlog(x)$. How to show it is strongly convex, i.e.,
Definition: Let $f: mathbb{R}^n rightarrow mathbb{R}$ be differentiable. Then $f$ is strongly convex if $exists$ a positive constant $alpha > 0$ such that
$$
langle nabla f(y) - nabla f(x),y-x rangle geq alpha||y-x||^2 ,,,,,,,,,forall x,y in mathbb{R}^n tag{1}
$$
Following $(1)$ we have $(y-x)log(frac{y}{x})geq alpha(y-x)^2
,,,,,
forall x,y in mathbb{R}_+$.
What $alpha$ satisfies the above and how we can handle the inequality $forall x,y in mathbb{R}_+$ ?
inequality logarithms convex-analysis
asked Jan 17 at 17:52
SaeedSaeed
1,005310
$endgroup$
$begingroup$
Why don't you calculate the second derivative?
$endgroup$
– Dr. Sonnhard Graubner
Jan 17 at 17:53
$begingroup$
@Dr. Sonnhard Graubner : Because I want to better understand the definition to proof the general case:math.stackexchange.com/questions/3055187/… and solve it.
$endgroup$
– Saeed
Jan 17 at 17:57
$begingroup$
@Dr. Sonnhard Graubner : Also, the second derivative does not specify $alpha$. Actually, I do not know how to use this fact saying strongly convex implies $nabla^2f(x) succeq alpha I$ for this case to get $alpha$.
$endgroup$
– Saeed
Jan 17 at 18:01
$begingroup$
The case $x=1,,y=1+z>0$ gives $frac{ln(1+z)}{z}gealpha$, which for sufficiently large $z>0$ contradicts any proposed $alpha>0$.
$endgroup$
– J.G.
Jan 17 at 18:27
$begingroup$
@J.G. : I think I should have assumed that $0 leq x leq 1$?
$endgroup$
– Saeed
Jan 17 at 18:32
|
show 1 more comment
$begingroup$
Let $f: mathbb{R}_+ rightarrow mathbb{R}$ where $f(x)=xlog(x)$. How to show it is strongly convex, i.e.,
Definition: Let $f: mathbb{R}^n rightarrow mathbb{R}$ be differentiable. Then $f$ is strongly convex if $exists$ a positive constant $alpha > 0$ such that
$$
langle nabla f(y) - nabla f(x),y-x rangle geq alpha||y-x||^2 ,,,,,,,,,forall x,y in mathbb{R}^n tag{1}
$$
Following $(1)$ we have $(y-x)log(frac{y}{x})geq alpha(y-x)^2
,,,,,
forall x,y in mathbb{R}_+$.
What $alpha$ satisfies the above and how we can handle the inequality $forall x,y in mathbb{R}_+$ ?
inequality logarithms convex-analysis
asked Jan 17 at 17:52
SaeedSaeed
1,005310
$endgroup$
Let $f: mathbb{R}_+ rightarrow mathbb{R}$ where $f(x)=xlog(x)$. How to show it is strongly convex, i.e.,
Definition: Let $f: mathbb{R}^n rightarrow mathbb{R}$ be differentiable. Then $f$ is strongly convex if $exists$ a positive constant $alpha > 0$ such that
$$
langle nabla f(y) - nabla f(x),y-x rangle geq alpha||y-x||^2 ,,,,,,,,,forall x,y in mathbb{R}^n tag{1}
$$
Following $(1)$ we have $(y-x)log(frac{y}{x})geq alpha(y-x)^2
,,,,,
forall x,y in mathbb{R}_+$.
What $alpha$ satisfies the above and how we can handle the inequality $forall x,y in mathbb{R}_+$ ?
inequality logarithms convex-analysis
inequality logarithms convex-analysis
asked Jan 17 at 17:52
SaeedSaeed
1,005310
asked Jan 17 at 17:52
SaeedSaeed
1,005310
asked Jan 17 at 17:52
SaeedSaeed
1,005310
asked Jan 17 at 17:52
SaeedSaeed
1,005310
asked Jan 17 at 17:52
SaeedSaeed
1,005310
1,005310
$begingroup$
Why don't you calculate the second derivative?
$endgroup$
– Dr. Sonnhard Graubner
Jan 17 at 17:53
$begingroup$
@Dr. Sonnhard Graubner : Because I want to better understand the definition to proof the general case:math.stackexchange.com/questions/3055187/… and solve it.
$endgroup$
– Saeed
Jan 17 at 17:57
$begingroup$
@Dr. Sonnhard Graubner : Also, the second derivative does not specify $alpha$. Actually, I do not know how to use this fact saying strongly convex implies $nabla^2f(x) succeq alpha I$ for this case to get $alpha$.
$endgroup$
– Saeed
Jan 17 at 18:01
$begingroup$
The case $x=1,,y=1+z>0$ gives $frac{ln(1+z)}{z}gealpha$, which for sufficiently large $z>0$ contradicts any proposed $alpha>0$.
$endgroup$
– J.G.
Jan 17 at 18:27
$begingroup$
@J.G. : I think I should have assumed that $0 leq x leq 1$?
$endgroup$
– Saeed
Jan 17 at 18:32
|
show 1 more comment
$begingroup$
Why don't you calculate the second derivative?
$endgroup$
– Dr. Sonnhard Graubner
Jan 17 at 17:53
$begingroup$
@Dr. Sonnhard Graubner : Because I want to better understand the definition to proof the general case:math.stackexchange.com/questions/3055187/… and solve it.
$endgroup$
– Saeed
Jan 17 at 17:57
$begingroup$
@Dr. Sonnhard Graubner : Also, the second derivative does not specify $alpha$. Actually, I do not know how to use this fact saying strongly convex implies $nabla^2f(x) succeq alpha I$ for this case to get $alpha$.
$endgroup$
– Saeed
Jan 17 at 18:01
$begingroup$
The case $x=1,,y=1+z>0$ gives $frac{ln(1+z)}{z}gealpha$, which for sufficiently large $z>0$ contradicts any proposed $alpha>0$.
$endgroup$
– J.G.
Jan 17 at 18:27
$begingroup$
@J.G. : I think I should have assumed that $0 leq x leq 1$?
$endgroup$
– Saeed
Jan 17 at 18:32
$begingroup$
Why don't you calculate the second derivative?
$endgroup$
– Dr. Sonnhard Graubner
Jan 17 at 17:53
$begingroup$
Why don't you calculate the second derivative?
$endgroup$
– Dr. Sonnhard Graubner
Jan 17 at 17:53
$begingroup$
@Dr. Sonnhard Graubner : Because I want to better understand the definition to proof the general case:math.stackexchange.com/questions/3055187/… and solve it.
$endgroup$
– Saeed
Jan 17 at 17:57
$begingroup$
@Dr. Sonnhard Graubner : Because I want to better understand the definition to proof the general case:math.stackexchange.com/questions/3055187/… and solve it.
$endgroup$
– Saeed
Jan 17 at 17:57
$begingroup$
@Dr. Sonnhard Graubner : Also, the second derivative does not specify $alpha$. Actually, I do not know how to use this fact saying strongly convex implies $nabla^2f(x) succeq alpha I$ for this case to get $alpha$.
$endgroup$
– Saeed
Jan 17 at 18:01
$begingroup$
@Dr. Sonnhard Graubner : Also, the second derivative does not specify $alpha$. Actually, I do not know how to use this fact saying strongly convex implies $nabla^2f(x) succeq alpha I$ for this case to get $alpha$.
$endgroup$
– Saeed
Jan 17 at 18:01
$begingroup$
The case $x=1,,y=1+z>0$ gives $frac{ln(1+z)}{z}gealpha$, which for sufficiently large $z>0$ contradicts any proposed $alpha>0$.
$endgroup$
– J.G.
Jan 17 at 18:27
$begingroup$
The case $x=1,,y=1+z>0$ gives $frac{ln(1+z)}{z}gealpha$, which for sufficiently large $z>0$ contradicts any proposed $alpha>0$.
$endgroup$
– J.G.
Jan 17 at 18:27
$begingroup$
@J.G. : I think I should have assumed that $0 leq x leq 1$?
$endgroup$
– Saeed
Jan 17 at 18:32
$begingroup$
@J.G. : I think I should have assumed that $0 leq x leq 1$?
$endgroup$
– Saeed
Jan 17 at 18:32
|
show 1 more comment
1 Answer
1
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$begingroup$
We have $f'(x)=1+log(x)$. Strict convexity therefore means that there exists a strictly positive $alpha$ such that
$$[log(y)-log(x)](y-x)geqalpha(y-x)^2$$
holds. Without loss of generality I assume $y>x$. Then the above inequality requires that
$$log(y)-log(x)geqalpha(y-x).$$
Although you don't state it, I assume that the variables $x$ and $y$ live on $(0,1)$ (because they are probabilities). Since the slope of the $log$ function on the interval $(0,1)$ is larger than or equal to 1, you can choose any positive $alpha$ smaller than 1.
If $x$ and $y$ live on a general bounded interval $(0,M)$, then the argument goes through with $alpha<1/M$.
answered Jan 17 at 18:03
Gerhard S.Gerhard S.
1,05029
$endgroup$
$begingroup$
I understand the argument justifying $log(y)-log(x)geqalpha(y-x)$ but could you show it algebraically?
$endgroup$
– Saeed
Jan 17 at 18:53
$begingroup$
@Saeed: $log$ is concave. Hence, $log(x)leqlog(y)+(1/y)(x-y)$. Since $1/ygeq1/M$, one gets $log(y)-log(x)geq(1/y)(y-x)geq(1/M)(y-x)$.
$endgroup$
– Gerhard S.
Jan 17 at 19:27
$begingroup$
Smart way of showing that.
$endgroup$
– Saeed
Jan 17 at 19:32
add a comment |
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1 Answer
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$begingroup$
We have $f'(x)=1+log(x)$. Strict convexity therefore means that there exists a strictly positive $alpha$ such that
$$[log(y)-log(x)](y-x)geqalpha(y-x)^2$$
holds. Without loss of generality I assume $y>x$. Then the above inequality requires that
$$log(y)-log(x)geqalpha(y-x).$$
Although you don't state it, I assume that the variables $x$ and $y$ live on $(0,1)$ (because they are probabilities). Since the slope of the $log$ function on the interval $(0,1)$ is larger than or equal to 1, you can choose any positive $alpha$ smaller than 1.
If $x$ and $y$ live on a general bounded interval $(0,M)$, then the argument goes through with $alpha<1/M$.
answered Jan 17 at 18:03
Gerhard S.Gerhard S.
1,05029
$endgroup$
$begingroup$
I understand the argument justifying $log(y)-log(x)geqalpha(y-x)$ but could you show it algebraically?
$endgroup$
– Saeed
Jan 17 at 18:53
$begingroup$
@Saeed: $log$ is concave. Hence, $log(x)leqlog(y)+(1/y)(x-y)$. Since $1/ygeq1/M$, one gets $log(y)-log(x)geq(1/y)(y-x)geq(1/M)(y-x)$.
$endgroup$
– Gerhard S.
Jan 17 at 19:27
$begingroup$
Smart way of showing that.
$endgroup$
– Saeed
Jan 17 at 19:32
add a comment |
$begingroup$
We have $f'(x)=1+log(x)$. Strict convexity therefore means that there exists a strictly positive $alpha$ such that
$$[log(y)-log(x)](y-x)geqalpha(y-x)^2$$
holds. Without loss of generality I assume $y>x$. Then the above inequality requires that
$$log(y)-log(x)geqalpha(y-x).$$
Although you don't state it, I assume that the variables $x$ and $y$ live on $(0,1)$ (because they are probabilities). Since the slope of the $log$ function on the interval $(0,1)$ is larger than or equal to 1, you can choose any positive $alpha$ smaller than 1.
If $x$ and $y$ live on a general bounded interval $(0,M)$, then the argument goes through with $alpha<1/M$.
answered Jan 17 at 18:03
Gerhard S.Gerhard S.
1,05029
$endgroup$
$begingroup$
I understand the argument justifying $log(y)-log(x)geqalpha(y-x)$ but could you show it algebraically?
$endgroup$
– Saeed
Jan 17 at 18:53
$begingroup$
@Saeed: $log$ is concave. Hence, $log(x)leqlog(y)+(1/y)(x-y)$. Since $1/ygeq1/M$, one gets $log(y)-log(x)geq(1/y)(y-x)geq(1/M)(y-x)$.
$endgroup$
– Gerhard S.
Jan 17 at 19:27
$begingroup$
Smart way of showing that.
$endgroup$
– Saeed
Jan 17 at 19:32
add a comment |
$begingroup$
We have $f'(x)=1+log(x)$. Strict convexity therefore means that there exists a strictly positive $alpha$ such that
$$[log(y)-log(x)](y-x)geqalpha(y-x)^2$$
holds. Without loss of generality I assume $y>x$. Then the above inequality requires that
$$log(y)-log(x)geqalpha(y-x).$$
Although you don't state it, I assume that the variables $x$ and $y$ live on $(0,1)$ (because they are probabilities). Since the slope of the $log$ function on the interval $(0,1)$ is larger than or equal to 1, you can choose any positive $alpha$ smaller than 1.
If $x$ and $y$ live on a general bounded interval $(0,M)$, then the argument goes through with $alpha<1/M$.
answered Jan 17 at 18:03
Gerhard S.Gerhard S.
1,05029
$endgroup$
We have $f'(x)=1+log(x)$. Strict convexity therefore means that there exists a strictly positive $alpha$ such that
$$[log(y)-log(x)](y-x)geqalpha(y-x)^2$$
holds. Without loss of generality I assume $y>x$. Then the above inequality requires that
$$log(y)-log(x)geqalpha(y-x).$$
Although you don't state it, I assume that the variables $x$ and $y$ live on $(0,1)$ (because they are probabilities). Since the slope of the $log$ function on the interval $(0,1)$ is larger than or equal to 1, you can choose any positive $alpha$ smaller than 1.
If $x$ and $y$ live on a general bounded interval $(0,M)$, then the argument goes through with $alpha<1/M$.
answered Jan 17 at 18:03
Gerhard S.Gerhard S.
1,05029
edited Jan 17 at 18:08
answered Jan 17 at 18:03
Gerhard S.Gerhard S.
1,05029
answered Jan 17 at 18:03
Gerhard S.Gerhard S.
1,05029
answered Jan 17 at 18:03
Gerhard S.Gerhard S.
1,05029
1,05029
$begingroup$
I understand the argument justifying $log(y)-log(x)geqalpha(y-x)$ but could you show it algebraically?
$endgroup$
– Saeed
Jan 17 at 18:53
$begingroup$
@Saeed: $log$ is concave. Hence, $log(x)leqlog(y)+(1/y)(x-y)$. Since $1/ygeq1/M$, one gets $log(y)-log(x)geq(1/y)(y-x)geq(1/M)(y-x)$.
$endgroup$
– Gerhard S.
Jan 17 at 19:27
$begingroup$
Smart way of showing that.
$endgroup$
– Saeed
Jan 17 at 19:32
add a comment |
$begingroup$
I understand the argument justifying $log(y)-log(x)geqalpha(y-x)$ but could you show it algebraically?
$endgroup$
– Saeed
Jan 17 at 18:53
$begingroup$
@Saeed: $log$ is concave. Hence, $log(x)leqlog(y)+(1/y)(x-y)$. Since $1/ygeq1/M$, one gets $log(y)-log(x)geq(1/y)(y-x)geq(1/M)(y-x)$.
$endgroup$
– Gerhard S.
Jan 17 at 19:27
$begingroup$
Smart way of showing that.
$endgroup$
– Saeed
Jan 17 at 19:32
$begingroup$
I understand the argument justifying $log(y)-log(x)geqalpha(y-x)$ but could you show it algebraically?
$endgroup$
– Saeed
Jan 17 at 18:53
$begingroup$
I understand the argument justifying $log(y)-log(x)geqalpha(y-x)$ but could you show it algebraically?
$endgroup$
– Saeed
Jan 17 at 18:53
$begingroup$
@Saeed: $log$ is concave. Hence, $log(x)leqlog(y)+(1/y)(x-y)$. Since $1/ygeq1/M$, one gets $log(y)-log(x)geq(1/y)(y-x)geq(1/M)(y-x)$.
$endgroup$
– Gerhard S.
Jan 17 at 19:27
$begingroup$
@Saeed: $log$ is concave. Hence, $log(x)leqlog(y)+(1/y)(x-y)$. Since $1/ygeq1/M$, one gets $log(y)-log(x)geq(1/y)(y-x)geq(1/M)(y-x)$.
$endgroup$
– Gerhard S.
Jan 17 at 19:27
$begingroup$
Smart way of showing that.
$endgroup$
– Saeed
Jan 17 at 19:32
$begingroup$
Smart way of showing that.
$endgroup$
– Saeed
Jan 17 at 19:32
add a comment |
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$begingroup$
Why don't you calculate the second derivative?
$endgroup$
– Dr. Sonnhard Graubner
Jan 17 at 17:53
$begingroup$
@Dr. Sonnhard Graubner : Because I want to better understand the definition to proof the general case:math.stackexchange.com/questions/3055187/… and solve it.
$endgroup$
– Saeed
Jan 17 at 17:57
$begingroup$
@Dr. Sonnhard Graubner : Also, the second derivative does not specify $alpha$. Actually, I do not know how to use this fact saying strongly convex implies $nabla^2f(x) succeq alpha I$ for this case to get $alpha$.
$endgroup$
– Saeed
Jan 17 at 18:01
$begingroup$
The case $x=1,,y=1+z>0$ gives $frac{ln(1+z)}{z}gealpha$, which for sufficiently large $z>0$ contradicts any proposed $alpha>0$.
$endgroup$
– J.G.
Jan 17 at 18:27
$begingroup$
@J.G. : I think I should have assumed that $0 leq x leq 1$?
$endgroup$
– Saeed
Jan 17 at 18:32