Simple example of a function that is convex but not jointly convex?
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Can a function be convex is each of it's arguments but not jointly convex? If yes, please provide an example and if not, could you please provide a proof?
The definition of joint convexity I use is: For any function $f(x,y)$, we have $f(lambda x_1 + (1-lambda)x_2, lambda y_1 + (1-lambda)y_2) leq lambda f(x_1, y_1) + (1-lambda)f(x_2,y_2)$.
Convex in each argument implies $f(lambda x_1 + (1-lambda)x_2, y) leq lambda f(x_1, y) + (1-lambda)f(x_2, y)$ and similar for $y$
Also, please do include if it makes sense to take a strict inequalities in the convexity definition as well. Thank you.
functions convex-analysis
asked Jan 5 at 18:42
user1936752user1936752
5321513
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add a comment |
$begingroup$
Can a function be convex is each of it's arguments but not jointly convex? If yes, please provide an example and if not, could you please provide a proof?
The definition of joint convexity I use is: For any function $f(x,y)$, we have $f(lambda x_1 + (1-lambda)x_2, lambda y_1 + (1-lambda)y_2) leq lambda f(x_1, y_1) + (1-lambda)f(x_2,y_2)$.
Convex in each argument implies $f(lambda x_1 + (1-lambda)x_2, y) leq lambda f(x_1, y) + (1-lambda)f(x_2, y)$ and similar for $y$
Also, please do include if it makes sense to take a strict inequalities in the convexity definition as well. Thank you.
functions convex-analysis
asked Jan 5 at 18:42
user1936752user1936752
5321513
$endgroup$
4
$begingroup$
What about $f(x,y)=xy$?
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– A.Γ.
Jan 5 at 18:47
1
$begingroup$
What is a jointly convex function?
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– Mostafa Ayaz
Jan 5 at 18:50
add a comment |
$begingroup$
Can a function be convex is each of it's arguments but not jointly convex? If yes, please provide an example and if not, could you please provide a proof?
The definition of joint convexity I use is: For any function $f(x,y)$, we have $f(lambda x_1 + (1-lambda)x_2, lambda y_1 + (1-lambda)y_2) leq lambda f(x_1, y_1) + (1-lambda)f(x_2,y_2)$.
Convex in each argument implies $f(lambda x_1 + (1-lambda)x_2, y) leq lambda f(x_1, y) + (1-lambda)f(x_2, y)$ and similar for $y$
Also, please do include if it makes sense to take a strict inequalities in the convexity definition as well. Thank you.
functions convex-analysis
asked Jan 5 at 18:42
user1936752user1936752
5321513
$endgroup$
Can a function be convex is each of it's arguments but not jointly convex? If yes, please provide an example and if not, could you please provide a proof?
The definition of joint convexity I use is: For any function $f(x,y)$, we have $f(lambda x_1 + (1-lambda)x_2, lambda y_1 + (1-lambda)y_2) leq lambda f(x_1, y_1) + (1-lambda)f(x_2,y_2)$.
Convex in each argument implies $f(lambda x_1 + (1-lambda)x_2, y) leq lambda f(x_1, y) + (1-lambda)f(x_2, y)$ and similar for $y$
Also, please do include if it makes sense to take a strict inequalities in the convexity definition as well. Thank you.
functions convex-analysis
functions convex-analysis
asked Jan 5 at 18:42
user1936752user1936752
5321513
asked Jan 5 at 18:42
user1936752user1936752
5321513
edited Jan 5 at 19:28
user1936752
asked Jan 5 at 18:42
user1936752user1936752
5321513
asked Jan 5 at 18:42
user1936752user1936752
5321513
asked Jan 5 at 18:42
user1936752user1936752
5321513
5321513
4
$begingroup$
What about $f(x,y)=xy$?
$endgroup$
– A.Γ.
Jan 5 at 18:47
1
$begingroup$
What is a jointly convex function?
$endgroup$
– Mostafa Ayaz
Jan 5 at 18:50
add a comment |
4
$begingroup$
What about $f(x,y)=xy$?
$endgroup$
– A.Γ.
Jan 5 at 18:47
1
$begingroup$
What is a jointly convex function?
$endgroup$
– Mostafa Ayaz
Jan 5 at 18:50
4
4
$begingroup$
What about $f(x,y)=xy$?
$endgroup$
– A.Γ.
Jan 5 at 18:47
$begingroup$
What about $f(x,y)=xy$?
$endgroup$
– A.Γ.
Jan 5 at 18:47
1
1
$begingroup$
What is a jointly convex function?
$endgroup$
– Mostafa Ayaz
Jan 5 at 18:50
$begingroup$
What is a jointly convex function?
$endgroup$
– Mostafa Ayaz
Jan 5 at 18:50
add a comment |
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4
$begingroup$
What about $f(x,y)=xy$?
$endgroup$
– A.Γ.
Jan 5 at 18:47
1
$begingroup$
What is a jointly convex function?
$endgroup$
– Mostafa Ayaz
Jan 5 at 18:50