Mixing
convexity of the function $f(x,y)=x^4-alpha y^2$
$begingroup$
To study the convexity of this function I calculate the Hessian:
$$begin{bmatrix}12x^2 & 0 \ 0 & -2alpha end{bmatrix}$$
For $alpha<0$ the matrix is semidefinite-positive so the function is convex?
real-analysis
asked Jan 15 at 20:01
Giulia B.Giulia B.
505311
$endgroup$
add a comment |
$begingroup$
To study the convexity of this function I calculate the Hessian:
$$begin{bmatrix}12x^2 & 0 \ 0 & -2alpha end{bmatrix}$$
For $alpha<0$ the matrix is semidefinite-positive so the function is convex?
real-analysis
asked Jan 15 at 20:01
Giulia B.Giulia B.
505311
$endgroup$
add a comment |
$begingroup$
To study the convexity of this function I calculate the Hessian:
$$begin{bmatrix}12x^2 & 0 \ 0 & -2alpha end{bmatrix}$$
For $alpha<0$ the matrix is semidefinite-positive so the function is convex?
real-analysis
asked Jan 15 at 20:01
Giulia B.Giulia B.
505311
$endgroup$
To study the convexity of this function I calculate the Hessian:
$$begin{bmatrix}12x^2 & 0 \ 0 & -2alpha end{bmatrix}$$
For $alpha<0$ the matrix is semidefinite-positive so the function is convex?
real-analysis
real-analysis
asked Jan 15 at 20:01
Giulia B.Giulia B.
505311
asked Jan 15 at 20:01
Giulia B.Giulia B.
505311
asked Jan 15 at 20:01
Giulia B.Giulia B.
505311
asked Jan 15 at 20:01
Giulia B.Giulia B.
505311
asked Jan 15 at 20:01
Giulia B.Giulia B.
505311
505311
add a comment |
add a comment |
1 Answer
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$begingroup$
As a matter of fact, this function is strictly convex if $alpha$ is negative. Both $x^4$ and $-alpha y^2$ are strictly convex and, hence, their sum is also strictly convex.
answered Jan 15 at 20:06
Gerhard S.Gerhard S.
1,06529
$endgroup$
add a comment |
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1 Answer
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1 Answer
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$begingroup$
As a matter of fact, this function is strictly convex if $alpha$ is negative. Both $x^4$ and $-alpha y^2$ are strictly convex and, hence, their sum is also strictly convex.
answered Jan 15 at 20:06
Gerhard S.Gerhard S.
1,06529
$endgroup$
add a comment |
$begingroup$
As a matter of fact, this function is strictly convex if $alpha$ is negative. Both $x^4$ and $-alpha y^2$ are strictly convex and, hence, their sum is also strictly convex.
answered Jan 15 at 20:06
Gerhard S.Gerhard S.
1,06529
$endgroup$
add a comment |
$begingroup$
As a matter of fact, this function is strictly convex if $alpha$ is negative. Both $x^4$ and $-alpha y^2$ are strictly convex and, hence, their sum is also strictly convex.
answered Jan 15 at 20:06
Gerhard S.Gerhard S.
1,06529
$endgroup$
As a matter of fact, this function is strictly convex if $alpha$ is negative. Both $x^4$ and $-alpha y^2$ are strictly convex and, hence, their sum is also strictly convex.
answered Jan 15 at 20:06
Gerhard S.Gerhard S.
1,06529
answered Jan 15 at 20:06
Gerhard S.Gerhard S.
1,06529
answered Jan 15 at 20:06
Gerhard S.Gerhard S.
1,06529
answered Jan 15 at 20:06
Gerhard S.Gerhard S.
1,06529
1,06529
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