Optimum of coordinate-wise convex function
$begingroup$
Let $f colon mathbb{R}^n to mathbb{R}$ be a function that is convex in each coordinate:
$$
forall i in {1, dots, n}. forall x_1. dots forall x_{i-1}, forall x_{i+1}dots forall x_n. f(x_1,dots,x_{i-1},cdot,x_{i+1},dots, x_n) text{ is convex}
$$
Can $f$ have a local optimum that is not also a global optimum?
I already know that $f$ is not necessarily convex (consider, e.g., $f(x,y)=x cdot y$). Therefore, I suspect that in general, $f$ may have local optimas. However, I could not find a counterexample so far ($x cdot y$ has no optimum).
optimization convex-analysis convex-optimization maxima-minima
asked Jan 10 at 10:41
PeterPeter
363114
$endgroup$
add a comment |
$begingroup$
Let $f colon mathbb{R}^n to mathbb{R}$ be a function that is convex in each coordinate:
$$
forall i in {1, dots, n}. forall x_1. dots forall x_{i-1}, forall x_{i+1}dots forall x_n. f(x_1,dots,x_{i-1},cdot,x_{i+1},dots, x_n) text{ is convex}
$$
Can $f$ have a local optimum that is not also a global optimum?
I already know that $f$ is not necessarily convex (consider, e.g., $f(x,y)=x cdot y$). Therefore, I suspect that in general, $f$ may have local optimas. However, I could not find a counterexample so far ($x cdot y$ has no optimum).
optimization convex-analysis convex-optimization maxima-minima
asked Jan 10 at 10:41
PeterPeter
363114
$endgroup$
add a comment |
$begingroup$
Let $f colon mathbb{R}^n to mathbb{R}$ be a function that is convex in each coordinate:
$$
forall i in {1, dots, n}. forall x_1. dots forall x_{i-1}, forall x_{i+1}dots forall x_n. f(x_1,dots,x_{i-1},cdot,x_{i+1},dots, x_n) text{ is convex}
$$
Can $f$ have a local optimum that is not also a global optimum?
I already know that $f$ is not necessarily convex (consider, e.g., $f(x,y)=x cdot y$). Therefore, I suspect that in general, $f$ may have local optimas. However, I could not find a counterexample so far ($x cdot y$ has no optimum).
optimization convex-analysis convex-optimization maxima-minima
asked Jan 10 at 10:41
PeterPeter
363114
$endgroup$
Let $f colon mathbb{R}^n to mathbb{R}$ be a function that is convex in each coordinate:
$$
forall i in {1, dots, n}. forall x_1. dots forall x_{i-1}, forall x_{i+1}dots forall x_n. f(x_1,dots,x_{i-1},cdot,x_{i+1},dots, x_n) text{ is convex}
$$
Can $f$ have a local optimum that is not also a global optimum?
I already know that $f$ is not necessarily convex (consider, e.g., $f(x,y)=x cdot y$). Therefore, I suspect that in general, $f$ may have local optimas. However, I could not find a counterexample so far ($x cdot y$ has no optimum).
optimization convex-analysis convex-optimization maxima-minima
optimization convex-analysis convex-optimization maxima-minima
asked Jan 10 at 10:41
PeterPeter
363114
asked Jan 10 at 10:41
PeterPeter
363114
asked Jan 10 at 10:41
PeterPeter
363114
asked Jan 10 at 10:41
PeterPeter
363114
asked Jan 10 at 10:41
PeterPeter
363114
363114
add a comment |
add a comment |
1 Answer
1
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votes
$begingroup$
The function
$$
f(x,y)
=
x^2 , y^2 + 20 , x , y + x^2 + y^2 + x
$$
has two local minimizers (near $(-3,3)$ and $(3,-3)$) with different function values. Only one of these minimizers is the global minimizer.
answered Jan 11 at 8:36
gerwgerw
20k11334
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add a comment |
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1 Answer
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1 Answer
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active
oldest
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active
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active
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$begingroup$
The function
$$
f(x,y)
=
x^2 , y^2 + 20 , x , y + x^2 + y^2 + x
$$
has two local minimizers (near $(-3,3)$ and $(3,-3)$) with different function values. Only one of these minimizers is the global minimizer.
answered Jan 11 at 8:36
gerwgerw
20k11334
$endgroup$
add a comment |
$begingroup$
The function
$$
f(x,y)
=
x^2 , y^2 + 20 , x , y + x^2 + y^2 + x
$$
has two local minimizers (near $(-3,3)$ and $(3,-3)$) with different function values. Only one of these minimizers is the global minimizer.
answered Jan 11 at 8:36
gerwgerw
20k11334
$endgroup$
add a comment |
$begingroup$
The function
$$
f(x,y)
=
x^2 , y^2 + 20 , x , y + x^2 + y^2 + x
$$
has two local minimizers (near $(-3,3)$ and $(3,-3)$) with different function values. Only one of these minimizers is the global minimizer.
answered Jan 11 at 8:36
gerwgerw
20k11334
$endgroup$
The function
$$
f(x,y)
=
x^2 , y^2 + 20 , x , y + x^2 + y^2 + x
$$
has two local minimizers (near $(-3,3)$ and $(3,-3)$) with different function values. Only one of these minimizers is the global minimizer.
answered Jan 11 at 8:36
gerwgerw
20k11334
answered Jan 11 at 8:36
gerwgerw
20k11334
answered Jan 11 at 8:36
gerwgerw
20k11334
answered Jan 11 at 8:36
gerwgerw
20k11334
20k11334
add a comment |
add a comment |
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