Mixing
What is the main characteristics of separable and non-separable functions?
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I have read an article that says, from the optimization point of view, the separable functions are simpler then non-separable functions and optimization algorithms can find their optimal values more easily. What are the main characteristics of separable functions which cause such behavior?
optimization
asked May 18 '18 at 6:55
Payam AbdyPayam Abdy
807
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I have read an article that says, from the optimization point of view, the separable functions are simpler then non-separable functions and optimization algorithms can find their optimal values more easily. What are the main characteristics of separable functions which cause such behavior?
optimization
asked May 18 '18 at 6:55
Payam AbdyPayam Abdy
807
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add a comment |
$begingroup$
I have read an article that says, from the optimization point of view, the separable functions are simpler then non-separable functions and optimization algorithms can find their optimal values more easily. What are the main characteristics of separable functions which cause such behavior?
optimization
asked May 18 '18 at 6:55
Payam AbdyPayam Abdy
807
$endgroup$
I have read an article that says, from the optimization point of view, the separable functions are simpler then non-separable functions and optimization algorithms can find their optimal values more easily. What are the main characteristics of separable functions which cause such behavior?
optimization
optimization
asked May 18 '18 at 6:55
Payam AbdyPayam Abdy
807
asked May 18 '18 at 6:55
Payam AbdyPayam Abdy
807
asked May 18 '18 at 6:55
Payam AbdyPayam Abdy
807
asked May 18 '18 at 6:55
Payam AbdyPayam Abdy
807
asked May 18 '18 at 6:55
Payam AbdyPayam Abdy
807
807
add a comment |
add a comment |
1 Answer
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I have a partial knowledge on that subject but this may give you (very ?) few understandings.
Let us say that you want to minimize the function $f : mathbb{R}^{2} rightarrow mathbb{R}$ which can be written:
$$forall (x,y) in mathbb{R}^{2} ; f(x,y) = g(x) + h(y)$$ because it is separable.
Then, $min_{(x,y)}f(x,y) = min_{x}g(x) + min_{y}h(y)$, which transforms your initial problem on $mathbb{R}^{2}$ into two problems on $mathbb{R}$. This can be easier to solve.
You could also transform a complex problem of minimization along one variable into two simpler problems of minimization along two variables by simply adding an equality constraint:
$$min_{x}g(x) + h(Ax) = min_{(x,y); s.t ;y = Ax}g(x) + h(y)$$
answered Jan 25 at 8:50
AlbertoAlberto
333
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1 Answer
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1 Answer
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$begingroup$
I have a partial knowledge on that subject but this may give you (very ?) few understandings.
Let us say that you want to minimize the function $f : mathbb{R}^{2} rightarrow mathbb{R}$ which can be written:
$$forall (x,y) in mathbb{R}^{2} ; f(x,y) = g(x) + h(y)$$ because it is separable.
Then, $min_{(x,y)}f(x,y) = min_{x}g(x) + min_{y}h(y)$, which transforms your initial problem on $mathbb{R}^{2}$ into two problems on $mathbb{R}$. This can be easier to solve.
You could also transform a complex problem of minimization along one variable into two simpler problems of minimization along two variables by simply adding an equality constraint:
$$min_{x}g(x) + h(Ax) = min_{(x,y); s.t ;y = Ax}g(x) + h(y)$$
answered Jan 25 at 8:50
AlbertoAlberto
333
$endgroup$
add a comment |
$begingroup$
I have a partial knowledge on that subject but this may give you (very ?) few understandings.
Let us say that you want to minimize the function $f : mathbb{R}^{2} rightarrow mathbb{R}$ which can be written:
$$forall (x,y) in mathbb{R}^{2} ; f(x,y) = g(x) + h(y)$$ because it is separable.
Then, $min_{(x,y)}f(x,y) = min_{x}g(x) + min_{y}h(y)$, which transforms your initial problem on $mathbb{R}^{2}$ into two problems on $mathbb{R}$. This can be easier to solve.
You could also transform a complex problem of minimization along one variable into two simpler problems of minimization along two variables by simply adding an equality constraint:
$$min_{x}g(x) + h(Ax) = min_{(x,y); s.t ;y = Ax}g(x) + h(y)$$
answered Jan 25 at 8:50
AlbertoAlberto
333
$endgroup$
add a comment |
$begingroup$
I have a partial knowledge on that subject but this may give you (very ?) few understandings.
Let us say that you want to minimize the function $f : mathbb{R}^{2} rightarrow mathbb{R}$ which can be written:
$$forall (x,y) in mathbb{R}^{2} ; f(x,y) = g(x) + h(y)$$ because it is separable.
Then, $min_{(x,y)}f(x,y) = min_{x}g(x) + min_{y}h(y)$, which transforms your initial problem on $mathbb{R}^{2}$ into two problems on $mathbb{R}$. This can be easier to solve.
You could also transform a complex problem of minimization along one variable into two simpler problems of minimization along two variables by simply adding an equality constraint:
$$min_{x}g(x) + h(Ax) = min_{(x,y); s.t ;y = Ax}g(x) + h(y)$$
answered Jan 25 at 8:50
AlbertoAlberto
333
$endgroup$
I have a partial knowledge on that subject but this may give you (very ?) few understandings.
Let us say that you want to minimize the function $f : mathbb{R}^{2} rightarrow mathbb{R}$ which can be written:
$$forall (x,y) in mathbb{R}^{2} ; f(x,y) = g(x) + h(y)$$ because it is separable.
Then, $min_{(x,y)}f(x,y) = min_{x}g(x) + min_{y}h(y)$, which transforms your initial problem on $mathbb{R}^{2}$ into two problems on $mathbb{R}$. This can be easier to solve.
You could also transform a complex problem of minimization along one variable into two simpler problems of minimization along two variables by simply adding an equality constraint:
$$min_{x}g(x) + h(Ax) = min_{(x,y); s.t ;y = Ax}g(x) + h(y)$$
answered Jan 25 at 8:50
AlbertoAlberto
333
answered Jan 25 at 8:50
AlbertoAlberto
333
answered Jan 25 at 8:50
AlbertoAlberto
333
answered Jan 25 at 8:50
AlbertoAlberto
333
333
add a comment |
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