What's the terminology for this alternative minimization algorithm?The Remez AlgorithmAn eigenvalue algorithm to solve constrained quadratic form minimizationCan the Levenberg-Marquardt algorithm be used for minimization and not fittingHow to test convergence of an algorithm for constrained optimizationMinimization of The Blind Deconvolution FunctionalUnderstanding the conditions for which ADMM can be appliedPreconditioner for the GMRES method in the Uzawa algorithmObtaining a feasible solution for underdetermined system of linear equations satisfying inequality constraints
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What's the terminology for this alternative minimization algorithm?
The Remez AlgorithmAn eigenvalue algorithm to solve constrained quadratic form minimizationCan the Levenberg-Marquardt algorithm be used for minimization and not fittingHow to test convergence of an algorithm for constrained optimizationMinimization of The Blind Deconvolution FunctionalUnderstanding the conditions for which ADMM can be appliedPreconditioner for the GMRES method in the Uzawa algorithmObtaining a feasible solution for underdetermined system of linear equations satisfying inequality constraints
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Say the model is $F(x_1)G(x_2)Z(x_3) = y in mathbbR^N$, with $F,G,Z$ explicitly known, we are given observation of $y$ as $y_b in mathbbR^N$ to find the value of $x_1$, $x_2$, $x_3$ for each sample. Clearly this is an underdetermined inverse problem.
I have seen some iterative algorithm like this,
- start with random number of $x_1$, $x_2, x_3$.
- choose $x_1$, keep $x_2$, $x_3$ constant as change $x_1$ to $x_1^*$ such that
$$F(x_1^*)G(x_2)Z(x_3) = y_b$$
update $x_1 = x_1^*$
same thing for $x_2$, but fix $x_1$, $x_3$, and update $x_2 = x_2^*$.
repeat for $x_3$, and do the whole process several iterations until it doesn't change
final result is taken as the solution to the inverse problem.
I have seen this philosophy many times across different areas of computer science, just hope to know what is the big picture behind it. like ADMM?
optimization iterative-method admm
asked 8 hours ago
ArtificiallyIntelligenceArtificiallyIntelligence
1557 bronze badges
$endgroup$
add a comment |
$begingroup$
Say the model is $F(x_1)G(x_2)Z(x_3) = y in mathbbR^N$, with $F,G,Z$ explicitly known, we are given observation of $y$ as $y_b in mathbbR^N$ to find the value of $x_1$, $x_2$, $x_3$ for each sample. Clearly this is an underdetermined inverse problem.
I have seen some iterative algorithm like this,
- start with random number of $x_1$, $x_2, x_3$.
- choose $x_1$, keep $x_2$, $x_3$ constant as change $x_1$ to $x_1^*$ such that
$$F(x_1^*)G(x_2)Z(x_3) = y_b$$
update $x_1 = x_1^*$
same thing for $x_2$, but fix $x_1$, $x_3$, and update $x_2 = x_2^*$.
repeat for $x_3$, and do the whole process several iterations until it doesn't change
final result is taken as the solution to the inverse problem.
I have seen this philosophy many times across different areas of computer science, just hope to know what is the big picture behind it. like ADMM?
optimization iterative-method admm
asked 8 hours ago
ArtificiallyIntelligenceArtificiallyIntelligence
1557 bronze badges
$endgroup$
add a comment |
$begingroup$
Say the model is $F(x_1)G(x_2)Z(x_3) = y in mathbbR^N$, with $F,G,Z$ explicitly known, we are given observation of $y$ as $y_b in mathbbR^N$ to find the value of $x_1$, $x_2$, $x_3$ for each sample. Clearly this is an underdetermined inverse problem.
I have seen some iterative algorithm like this,
- start with random number of $x_1$, $x_2, x_3$.
- choose $x_1$, keep $x_2$, $x_3$ constant as change $x_1$ to $x_1^*$ such that
$$F(x_1^*)G(x_2)Z(x_3) = y_b$$
update $x_1 = x_1^*$
same thing for $x_2$, but fix $x_1$, $x_3$, and update $x_2 = x_2^*$.
repeat for $x_3$, and do the whole process several iterations until it doesn't change
final result is taken as the solution to the inverse problem.
I have seen this philosophy many times across different areas of computer science, just hope to know what is the big picture behind it. like ADMM?
optimization iterative-method admm
asked 8 hours ago
ArtificiallyIntelligenceArtificiallyIntelligence
1557 bronze badges
$endgroup$
Say the model is $F(x_1)G(x_2)Z(x_3) = y in mathbbR^N$, with $F,G,Z$ explicitly known, we are given observation of $y$ as $y_b in mathbbR^N$ to find the value of $x_1$, $x_2$, $x_3$ for each sample. Clearly this is an underdetermined inverse problem.
I have seen some iterative algorithm like this,
- start with random number of $x_1$, $x_2, x_3$.
- choose $x_1$, keep $x_2$, $x_3$ constant as change $x_1$ to $x_1^*$ such that
$$F(x_1^*)G(x_2)Z(x_3) = y_b$$
update $x_1 = x_1^*$
same thing for $x_2$, but fix $x_1$, $x_3$, and update $x_2 = x_2^*$.
repeat for $x_3$, and do the whole process several iterations until it doesn't change
final result is taken as the solution to the inverse problem.
I have seen this philosophy many times across different areas of computer science, just hope to know what is the big picture behind it. like ADMM?
optimization iterative-method admm
optimization iterative-method admm
asked 8 hours ago
ArtificiallyIntelligenceArtificiallyIntelligence
1557 bronze badges
asked 8 hours ago
ArtificiallyIntelligenceArtificiallyIntelligence
1557 bronze badges
asked 8 hours ago
ArtificiallyIntelligenceArtificiallyIntelligence
1557 bronze badges
asked 8 hours ago
ArtificiallyIntelligenceArtificiallyIntelligence
1557 bronze badges
asked 8 hours ago
ArtificiallyIntelligenceArtificiallyIntelligence
1557 bronze badges
1557 bronze badges
add a comment |
add a comment |
1 Answer
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A general name for this approach is "Block Coordinate Descent." It's important to understand that convergence isn't guaranteed without additional hypotheses.
ADMM is not simply block coordinate descent- it's a more complicated method that is optimizing with respect to primal variables $x$ and $z$ in each iteration and then adjusting the Lagrange multiplier and penalty parameter.
answered 7 hours ago
Brian BorchersBrian Borchers
13.9k1 gold badge22 silver badges53 bronze badges
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just found in a 1977 paper, it is called nonlinear Gauss Seidel method. oh yeah, now it looks more familiar.
$endgroup$
– ArtificiallyIntelligence
1 hour ago
add a comment |
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1 Answer
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1 Answer
1
active
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active
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votes
$begingroup$
A general name for this approach is "Block Coordinate Descent." It's important to understand that convergence isn't guaranteed without additional hypotheses.
ADMM is not simply block coordinate descent- it's a more complicated method that is optimizing with respect to primal variables $x$ and $z$ in each iteration and then adjusting the Lagrange multiplier and penalty parameter.
answered 7 hours ago
Brian BorchersBrian Borchers
13.9k1 gold badge22 silver badges53 bronze badges
$endgroup$
$begingroup$
just found in a 1977 paper, it is called nonlinear Gauss Seidel method. oh yeah, now it looks more familiar.
$endgroup$
– ArtificiallyIntelligence
1 hour ago
add a comment |
$begingroup$
A general name for this approach is "Block Coordinate Descent." It's important to understand that convergence isn't guaranteed without additional hypotheses.
ADMM is not simply block coordinate descent- it's a more complicated method that is optimizing with respect to primal variables $x$ and $z$ in each iteration and then adjusting the Lagrange multiplier and penalty parameter.
answered 7 hours ago
Brian BorchersBrian Borchers
13.9k1 gold badge22 silver badges53 bronze badges
$endgroup$
$begingroup$
just found in a 1977 paper, it is called nonlinear Gauss Seidel method. oh yeah, now it looks more familiar.
$endgroup$
– ArtificiallyIntelligence
1 hour ago
add a comment |
$begingroup$
A general name for this approach is "Block Coordinate Descent." It's important to understand that convergence isn't guaranteed without additional hypotheses.
ADMM is not simply block coordinate descent- it's a more complicated method that is optimizing with respect to primal variables $x$ and $z$ in each iteration and then adjusting the Lagrange multiplier and penalty parameter.
answered 7 hours ago
Brian BorchersBrian Borchers
13.9k1 gold badge22 silver badges53 bronze badges
$endgroup$
A general name for this approach is "Block Coordinate Descent." It's important to understand that convergence isn't guaranteed without additional hypotheses.
ADMM is not simply block coordinate descent- it's a more complicated method that is optimizing with respect to primal variables $x$ and $z$ in each iteration and then adjusting the Lagrange multiplier and penalty parameter.
answered 7 hours ago
Brian BorchersBrian Borchers
13.9k1 gold badge22 silver badges53 bronze badges
answered 7 hours ago
Brian BorchersBrian Borchers
13.9k1 gold badge22 silver badges53 bronze badges
answered 7 hours ago
Brian BorchersBrian Borchers
13.9k1 gold badge22 silver badges53 bronze badges
answered 7 hours ago
Brian BorchersBrian Borchers
13.9k1 gold badge22 silver badges53 bronze badges
13.9k1 gold badge22 silver badges53 bronze badges
$begingroup$
just found in a 1977 paper, it is called nonlinear Gauss Seidel method. oh yeah, now it looks more familiar.
$endgroup$
– ArtificiallyIntelligence
1 hour ago
add a comment |
$begingroup$
just found in a 1977 paper, it is called nonlinear Gauss Seidel method. oh yeah, now it looks more familiar.
$endgroup$
– ArtificiallyIntelligence
1 hour ago
$begingroup$
just found in a 1977 paper, it is called nonlinear Gauss Seidel method. oh yeah, now it looks more familiar.
$endgroup$
– ArtificiallyIntelligence
1 hour ago
$begingroup$
just found in a 1977 paper, it is called nonlinear Gauss Seidel method. oh yeah, now it looks more familiar.
$endgroup$
– ArtificiallyIntelligence
1 hour ago
add a comment |
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