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(Geometric and arithmetic mean)How to prove the function $G(x)-alpha A(x)$ is concave?
The 2019 Stack Overflow Developer Survey Results Are InIs the geometric-to-arithmetic function convex or concave?Concave function and its EigenvaluesProve domain of a concave function is convexIs $frac1xylog(xya+1)$ a convex function over the domain $x, y in (0,1], a>0$How is it a concave function?How to prove that its a convex function?Proving that $(x+y+2z)(2x+4y+z)(x+y+z)$ defines a convex level setProve Neg. Log Likelihood for Gaussian distribution is convex in mean and variance.How to find the following condition on the domain of conjugate function of negative geometric mean?Hessian of negative log-likelihood of logistic regression is positive definite?
$begingroup$
The geometric and arithmetic means of $xin R^n_+$ are, respectively,
$$G(x)=(prod^n_i=1x_i)^1/n,quad A(x)=frac1nsum^n_i=1x_i$$
Suppose $0leq alphaleq 1$, how to prove the function $G(x)-alpha A(x)$ is concave?
I try to solve it by Hessian matrix, but it seems not work.
convex-optimization
asked Mar 24 at 7:52
NoilynNoilyn
11
$endgroup$
add a comment |
$begingroup$
The geometric and arithmetic means of $xin R^n_+$ are, respectively,
$$G(x)=(prod^n_i=1x_i)^1/n,quad A(x)=frac1nsum^n_i=1x_i$$
Suppose $0leq alphaleq 1$, how to prove the function $G(x)-alpha A(x)$ is concave?
I try to solve it by Hessian matrix, but it seems not work.
convex-optimization
asked Mar 24 at 7:52
NoilynNoilyn
11
$endgroup$
$begingroup$
The geometric mean is not that you defined...
$endgroup$
– Mostafa Ayaz
Mar 24 at 7:57
$begingroup$
Sorry, I made a mistake.
$endgroup$
– Noilyn
Mar 24 at 8:06
1
$begingroup$
Seems quite easy. The geometric mean is concave. Any linear function, including $-alpha A(x)$ is concave. A sum of concave functions is concave. Am I missing something?
$endgroup$
– Alex Shtof
Mar 24 at 8:37
$begingroup$
@AlexShtof Yeah，you are right. Thankssss! (I'm so stupid😦)
$endgroup$
– Noilyn
Mar 24 at 8:40
add a comment |
$begingroup$
The geometric and arithmetic means of $xin R^n_+$ are, respectively,
$$G(x)=(prod^n_i=1x_i)^1/n,quad A(x)=frac1nsum^n_i=1x_i$$
Suppose $0leq alphaleq 1$, how to prove the function $G(x)-alpha A(x)$ is concave?
I try to solve it by Hessian matrix, but it seems not work.
convex-optimization
asked Mar 24 at 7:52
NoilynNoilyn
11
$endgroup$
The geometric and arithmetic means of $xin R^n_+$ are, respectively,
$$G(x)=(prod^n_i=1x_i)^1/n,quad A(x)=frac1nsum^n_i=1x_i$$
Suppose $0leq alphaleq 1$, how to prove the function $G(x)-alpha A(x)$ is concave?
I try to solve it by Hessian matrix, but it seems not work.
convex-optimization
convex-optimization
asked Mar 24 at 7:52
NoilynNoilyn
11
asked Mar 24 at 7:52
NoilynNoilyn
11
edited Mar 24 at 8:05
Noilyn
asked Mar 24 at 7:52
NoilynNoilyn
11
asked Mar 24 at 7:52
NoilynNoilyn
11
asked Mar 24 at 7:52
NoilynNoilyn
11
11
$begingroup$
The geometric mean is not that you defined...
$endgroup$
– Mostafa Ayaz
Mar 24 at 7:57
$begingroup$
Sorry, I made a mistake.
$endgroup$
– Noilyn
Mar 24 at 8:06
1
$begingroup$
Seems quite easy. The geometric mean is concave. Any linear function, including $-alpha A(x)$ is concave. A sum of concave functions is concave. Am I missing something?
$endgroup$
– Alex Shtof
Mar 24 at 8:37
$begingroup$
@AlexShtof Yeah，you are right. Thankssss! (I'm so stupid😦)
$endgroup$
– Noilyn
Mar 24 at 8:40
add a comment |
$begingroup$
The geometric mean is not that you defined...
$endgroup$
– Mostafa Ayaz
Mar 24 at 7:57
$begingroup$
Sorry, I made a mistake.
$endgroup$
– Noilyn
Mar 24 at 8:06
1
$begingroup$
Seems quite easy. The geometric mean is concave. Any linear function, including $-alpha A(x)$ is concave. A sum of concave functions is concave. Am I missing something?
$endgroup$
– Alex Shtof
Mar 24 at 8:37
$begingroup$
@AlexShtof Yeah，you are right. Thankssss! (I'm so stupid😦)
$endgroup$
– Noilyn
Mar 24 at 8:40
$begingroup$
The geometric mean is not that you defined...
$endgroup$
– Mostafa Ayaz
Mar 24 at 7:57
$begingroup$
The geometric mean is not that you defined...
$endgroup$
– Mostafa Ayaz
Mar 24 at 7:57
$begingroup$
Sorry, I made a mistake.
$endgroup$
– Noilyn
Mar 24 at 8:06
$begingroup$
Sorry, I made a mistake.
$endgroup$
– Noilyn
Mar 24 at 8:06
1
1
$begingroup$
Seems quite easy. The geometric mean is concave. Any linear function, including $-alpha A(x)$ is concave. A sum of concave functions is concave. Am I missing something?
$endgroup$
– Alex Shtof
Mar 24 at 8:37
$begingroup$
Seems quite easy. The geometric mean is concave. Any linear function, including $-alpha A(x)$ is concave. A sum of concave functions is concave. Am I missing something?
$endgroup$
– Alex Shtof
Mar 24 at 8:37
$begingroup$
@AlexShtof Yeah，you are right. Thankssss! (I'm so stupid😦)
$endgroup$
– Noilyn
Mar 24 at 8:40
$begingroup$
@AlexShtof Yeah，you are right. Thankssss! (I'm so stupid😦)
$endgroup$
– Noilyn
Mar 24 at 8:40
add a comment |
0
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$begingroup$
The geometric mean is not that you defined...
$endgroup$
– Mostafa Ayaz
Mar 24 at 7:57
$begingroup$
Sorry, I made a mistake.
$endgroup$
– Noilyn
Mar 24 at 8:06
1
$begingroup$
Seems quite easy. The geometric mean is concave. Any linear function, including $-alpha A(x)$ is concave. A sum of concave functions is concave. Am I missing something?
$endgroup$
– Alex Shtof
Mar 24 at 8:37
$begingroup$
@AlexShtof Yeah，you are right. Thankssss! (I'm so stupid😦)
$endgroup$
– Noilyn
Mar 24 at 8:40