Mixing
Convergence in multi-objective coordinate descent
$begingroup$
I have two functions $f_1(x;y)$ and $f_2(y;x)$, which are both optimized in terms of the same variables $x$ and $y$. The argument notation with semicolon (e.g. $x;y$ in $f_1(x;y)$) should demonstrate that when optimizing $f_1$, only $x$ can be adapted and $y$ is considered as fixed (symmetric for $f_2$).
The two functions $f_1$ and $f_2$ are:
- $f_1(x;y) = xcdot(x+y)^2 + (1-x)cdot(2-x-y)$
- $f_2(y;x) = ycdot(x+y)^2 + (1-y)cdot(2-x-y)$
An optimization round works as follows (a kind of coordinate descent with multiple objective functions):
- $x^{(k+1)} = argmin_{x} f_1(x;y^{(k)})$
- $y^{(k+1)} = argmin_{y} f_2(y;x^{(k+1)})$
My question: How can we prove (or disprove) the convergence of such an optimization algorithm, where convergence should yield $(x',y')$ s.t.
$x' = argmin_{x} f_1(x;y')$ and- $y' = argmin_{y} f_2(y;x')$
i.e. both functions $f_1$ and $f_2$ are at the minimum with respect to their adjustable variable, given the fixed variable.
(I am not interested in the global minima of $f_1$ or $f_2$.)
convergence optimization convex-optimization
asked Jan 3 at 8:17
simaschsimasch
13
$endgroup$
add a comment |
$begingroup$
I have two functions $f_1(x;y)$ and $f_2(y;x)$, which are both optimized in terms of the same variables $x$ and $y$. The argument notation with semicolon (e.g. $x;y$ in $f_1(x;y)$) should demonstrate that when optimizing $f_1$, only $x$ can be adapted and $y$ is considered as fixed (symmetric for $f_2$).
The two functions $f_1$ and $f_2$ are:
- $f_1(x;y) = xcdot(x+y)^2 + (1-x)cdot(2-x-y)$
- $f_2(y;x) = ycdot(x+y)^2 + (1-y)cdot(2-x-y)$
An optimization round works as follows (a kind of coordinate descent with multiple objective functions):
- $x^{(k+1)} = argmin_{x} f_1(x;y^{(k)})$
- $y^{(k+1)} = argmin_{y} f_2(y;x^{(k+1)})$
My question: How can we prove (or disprove) the convergence of such an optimization algorithm, where convergence should yield $(x',y')$ s.t.
$x' = argmin_{x} f_1(x;y')$ and- $y' = argmin_{y} f_2(y;x')$
i.e. both functions $f_1$ and $f_2$ are at the minimum with respect to their adjustable variable, given the fixed variable.
(I am not interested in the global minima of $f_1$ or $f_2$.)
convergence optimization convex-optimization
asked Jan 3 at 8:17
simaschsimasch
13
$endgroup$
add a comment |
$begingroup$
I have two functions $f_1(x;y)$ and $f_2(y;x)$, which are both optimized in terms of the same variables $x$ and $y$. The argument notation with semicolon (e.g. $x;y$ in $f_1(x;y)$) should demonstrate that when optimizing $f_1$, only $x$ can be adapted and $y$ is considered as fixed (symmetric for $f_2$).
The two functions $f_1$ and $f_2$ are:
- $f_1(x;y) = xcdot(x+y)^2 + (1-x)cdot(2-x-y)$
- $f_2(y;x) = ycdot(x+y)^2 + (1-y)cdot(2-x-y)$
An optimization round works as follows (a kind of coordinate descent with multiple objective functions):
- $x^{(k+1)} = argmin_{x} f_1(x;y^{(k)})$
- $y^{(k+1)} = argmin_{y} f_2(y;x^{(k+1)})$
My question: How can we prove (or disprove) the convergence of such an optimization algorithm, where convergence should yield $(x',y')$ s.t.
$x' = argmin_{x} f_1(x;y')$ and- $y' = argmin_{y} f_2(y;x')$
i.e. both functions $f_1$ and $f_2$ are at the minimum with respect to their adjustable variable, given the fixed variable.
(I am not interested in the global minima of $f_1$ or $f_2$.)
convergence optimization convex-optimization
asked Jan 3 at 8:17
simaschsimasch
13
$endgroup$
I have two functions $f_1(x;y)$ and $f_2(y;x)$, which are both optimized in terms of the same variables $x$ and $y$. The argument notation with semicolon (e.g. $x;y$ in $f_1(x;y)$) should demonstrate that when optimizing $f_1$, only $x$ can be adapted and $y$ is considered as fixed (symmetric for $f_2$).
The two functions $f_1$ and $f_2$ are:
- $f_1(x;y) = xcdot(x+y)^2 + (1-x)cdot(2-x-y)$
- $f_2(y;x) = ycdot(x+y)^2 + (1-y)cdot(2-x-y)$
An optimization round works as follows (a kind of coordinate descent with multiple objective functions):
- $x^{(k+1)} = argmin_{x} f_1(x;y^{(k)})$
- $y^{(k+1)} = argmin_{y} f_2(y;x^{(k+1)})$
My question: How can we prove (or disprove) the convergence of such an optimization algorithm, where convergence should yield $(x',y')$ s.t.
$x' = argmin_{x} f_1(x;y')$ and- $y' = argmin_{y} f_2(y;x')$
i.e. both functions $f_1$ and $f_2$ are at the minimum with respect to their adjustable variable, given the fixed variable.
(I am not interested in the global minima of $f_1$ or $f_2$.)
convergence optimization convex-optimization
convergence optimization convex-optimization
asked Jan 3 at 8:17
simaschsimasch
13
asked Jan 3 at 8:17
simaschsimasch
13
edited Jan 4 at 7:59
simasch
asked Jan 3 at 8:17
simaschsimasch
13
asked Jan 3 at 8:17
simaschsimasch
13
asked Jan 3 at 8:17
simaschsimasch
13
13
add a comment |
add a comment |
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