Simple example of a function that is convex but not jointly convex?
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 2 days ago
user1936752user1936752
5201513
add a comment |
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 2 days ago
user1936752user1936752
5201513
3
What about $f(x,y)=xy$?
– A.Γ.
2 days ago
What is a jointly convex function?
– Mostafa Ayaz
2 days ago
add a comment |
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 2 days ago
user1936752user1936752
5201513
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 2 days ago
user1936752user1936752
5201513
asked 2 days ago
user1936752user1936752
5201513
edited 2 days ago
user1936752
asked 2 days ago
user1936752user1936752
5201513
asked 2 days ago
user1936752user1936752
5201513
asked 2 days ago
user1936752user1936752
5201513
5201513
3
What about $f(x,y)=xy$?
– A.Γ.
2 days ago
What is a jointly convex function?
– Mostafa Ayaz
2 days ago
add a comment |
3
What about $f(x,y)=xy$?
– A.Γ.
2 days ago
What is a jointly convex function?
– Mostafa Ayaz
2 days ago
3
3
What about $f(x,y)=xy$?
– A.Γ.
2 days ago
What about $f(x,y)=xy$?
– A.Γ.
2 days ago
What is a jointly convex function?
– Mostafa Ayaz
2 days ago
What is a jointly convex function?
– Mostafa Ayaz
2 days ago
add a comment |
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3
What about $f(x,y)=xy$?
– A.Γ.
2 days ago
What is a jointly convex function?
– Mostafa Ayaz
2 days ago